On the logical strengths of partial solutions to mathematical problems

We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood [‘Reverse mathematics and a Ramsey‐type König's lemma’, J. Symb. Log. 77 (2012) 1272–1280], we say that a Ramsey‐type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey‐type variants of problems related to König's lemma, such as restrictions of König's lemma, Boolean satisfiability problems and graph coloring problems. We find that sometimes the Ramsey‐type variant of a problem is strictly easier than the original problem (as Flood showed with weak König's lemma) and that sometimes the Ramsey‐type variant of a problem is equivalent to the original problem. We show that the Ramsey‐type variant of weak König's lemma is robust in the sense of Montalbán [‘Open questions in reverse mathematics’, Bull. Symb. Log. 17 (2011) 431–454]: it is equivalent to several perturbations. We also clarify the relationship between Ramsey‐type weak König's lemma and algorithmic randomness by showing that Ramsey‐type weak weak König's lemma is equivalent to the problem of finding diagonally non‐recursive functions and that these problems are strictly easier than Ramsey‐type weak König's lemma. This answers a question of Flood.


INTRODUCTION 1.Ramsey-type mathematical theorems
Reverse mathematics is a foundational program whose aim is to classify the theorems of ordinary (i.e., non-set-theoretic) mathematics according to their provability strengths.These theorems are formalized in the language of second-order arithmetic (which suffices for most ordinary theorems, even those concerning the structure of the real line or analysis on complete separable metric spaces), and the implications among them are studied over a base theory called RCA 0 .Roughly speaking, the theorems provable in RCA 0 are those that are computable (or effective) in the sense illustrated by the example of the intermediate value theorem.Given a representation of a continuous realvalued function which is negative at 0 and positive at 1, from this representation one can compute an x ∈ (0, 1) such that f (x) = 0 using the usual interval-halving procedure.This procedure can only fail if the function is 0 at a rational, but then this rational itself is the desired x.In reverse mathematics, we say that the intermediate value theorem is provable in RCA 0 .
Other classical theorems are not computable.König's lemma, which states that every finitely branching tree with infinitely many nodes must have an infinite path, is an important example.If we are given a representation of a finitely branching infinite tree, we may not be able to compute a path through the tree from the representation.Indeed, there exist finitely branching infinite recursive trees without recursive paths.In reverse mathematics, we say that König's lemma is not provable in RCA 0 .
Implication over RCA 0 provides a natural classification of logical strength.We think of a theorem ϕ (formalized as a sentence of second-order arithmetic) as being at least as strong as a theorem ψ Received by the editors November 24, 2014.Paul Shafer is an FWO Pegasus Long Postdoctoral Fellow.He was also supported by the Fondation Sciences Mathématiques de Paris.Laurent Bienvenu and Ludovic Patey are funded by the John Templeton Foundation ('Structure and Randomness in the Theory of Computation' project).The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the John Templeton Foundation.
if ϕ → ψ can be proved in RCA 0 .Similarly, we think of ϕ and ψ as having equivalent strength if ϕ ↔ ψ can be proved in RCA 0 .A truly remarkable fact is that, under this classification, five equivalence classes, called the Big Five, emerge and classify the majority of usual theorems.There is, however, a notable family of theorems which do not fall into the Big Five classes.These are what we call the Ramsey-type theorems.Perhaps the most famous (or maybe infamous) Ramseytype theorem is the theorem from which 'Ramsey-type' gets its name: Ramsey's theorem for pairs.Ramsey's theorem for pairs states that if each pair of integers is colored one of some finite number of colors, then there is an infinite subset H of integers such that every pair of integers from H is colored the same color.Since the seminal paper of Cholak, Jockusch, and Slaman [4], an abundant literature developed surrounding the strength of this theorem and the strengths of related (usually weaker) theorems, such as chain-antichain, ascending or descending sequence, and the Erdős-Moser theorem (see, for example, [15] and [22]).
In general, we think of a theorem as being a Ramsey-type theorem if its solutions are closed under taking infinite subsets.Let us explain what we mean by this.Often a theorem can be interpreted as expressing that every instance of some mathematical problem has a solution.For example, with Ramsey's theorem for pairs, an instance of the corresponding problem is a coloring of pairs of integers using finitely many colors, and the solutions to this instance are the sets H satisfying the theorem's conclusion.Ramsey's theorem for pairs is a Ramsey-type theorem because if H is a solution to some instance of the corresponding problem, then every infinite H 0 ⊆ H is also a solution to this same instance.
König's lemma, on the other hand, corresponds to the problem whose instances are finitely branching, infinite trees and whose solutions are the paths through these trees.König's lemma has a satisfying computability-theoretic interpretation.There is a recursive, finitely branching, infinite tree, all of whose paths compute the Turing jump, and the Turing jump can compute a path through any given recursive, finitely branching, infinite tree.Thus König's lemma, as well as all the theorems in its equivalence class (called ACA 0 ), can be thought of as being equivalent to asserting that for every set X , the Turing jump of X exists.The other classes of the big five (RCA 0 , WKL 0 , ATR 0 , and Π 1 1 -CA 0 ) can also be characterized in computability-theoretic terms (respectively as closure under: Turing reducibility and Turing joins, the existence of PA-degrees, the existence of transfinite Turing jumps, and the existence of Turing hyperjumps).König's lemma is clearly not a Ramsey-type theorem.The Ramsey-type theorems do not typically have nice computability-theoretic characterizations of their equivalence classes (though some do: Ramsey's theorem for n-tuples for each fixed n ≥ 3 is in the ACA 0 class).There are very few computability-theoretic properties which are preserved under taking infinite subsets.
An interesting idea of Flood was to better understand Ramsey-type theorems by introducing Ramsey-type weak König's lemma (here denoted RWKL), which is a hybrid statement implied by both weak König's lemma and Ramsey's theorem for pairs.RWKL asserts that for every infinite binary tree, there is an infinite set that can be extended to a path or whose complement can be extended to path.RWKL is of course an artificial weakening of a natural theorem (König's lemma restricted to binary trees, which in turn is equivalent to many classical theorems).However, RWKL is a useful tool for studying Ramsey-type theorems, and it does exhibit interesting connections with computability theory and algorithmic randomness as demonstrated by Flood and (hopefully) by this paper.We continue the study of RWKL.We look at several combinatorial principles which are equivalent to weak König's lemma and study their Ramsey-type variants.That is, instead of asserting that every instance of the corresponding problem has a solution, we assert that every instance of the corresponding problem has an infinite partial solution.In many cases the Ramseytype variant of a theorem is strictly weaker than the original theorem, though in some cases a theorem is equivalent to its Ramsey-type variant.
The paper is organized as follows.In the next section, we review the basics of reverse mathematics, leading to Flood's principle RWKL.In Section 3, we study several Ramsey-type variants of full, bounded, weak, and weak weak König's lemma.The remainder of the paper focuses on Ramseytype variants of theorems equivalent to weak König's lemma.In Section 4, we study Ramsey-type variants of the compactness theorem for propositional logic.In Section 5, we study the Ramsey-type variants of graph coloring theorems.In Section 6, we prove several non-implications concerning concerning the Ramsey-type theorems and answer a question from Flood [9].

Basic notation
We follow the standard notation from computability theory.(Φ e ) e∈ is an effective list of all partial recursive functions.W e = dom(Φ e ) is the e th r.e.set.These relativize to any oracle X , and we denote the corresponding lists by (Φ X e ) e∈ and (W X e ) e∈ .Identify each k ∈ with the set {0, 1, . . ., k − 1}.For k ∈ ∪ { } and s ∈ , k s is the set of strings of length s over k, k <s is the set of strings of length < s over k, k < is the set of finite strings over k, and k is the set of infinite strings over k.The length of a finite string σ is denoted |σ|.For i ∈ and σ a finite or infinite string, σ(i) is the (i + 1) th bit of σ.For finite or infinite strings σ and τ, σ is a prefix of τ (written σ τ) if dom(σ) ⊆ dom(τ) and (∀i ∈ dom(σ))(σ(i) = τ(i)).For an n ∈ and a string (finite or infinite) σ of length ≥ n, σ ↾ n = 〈σ(0), σ(1), . . ., σ(n − 1)〉 is the initial segment of σ of length n.
A tree is a set T ⊆ < such that ∀σ∀τ(σ ∈ T ∧ τ σ → τ ∈ T ).If T is a tree and s ∈ , then T s is the set of strings in T of length s.An f ∈ is a path through a tree T if (∀n ∈ )( f ↾ n ∈ T ).
The set of paths through T is denoted [T ].
For k ∈ ∪ { }, the space k is topologized by viewing it as i∈ k, giving each copy of k the discrete topology, and giving the product the product topology.Basic open sets, also called cylinders, are sets of the form σ = { f ∈ k : f σ} for σ ∈ k < .Open sets are of the form σ∈W σ for W ⊆ k < .If the set W is an r.e.subset of k < , then σ∈W σ is said to be r.e.(or effectively) open.
We identify the space 2 of infinite binary sequences with ( ) by equating each subset of with its characteristic string as usual. 2 is compact, and its clopen sets are exactly the finite unions of cylinders.The uniform (or Lebesgue) measure µ on 2 is the Borel probability measure for which It is a convention, when working in second-order arithmetic, to use the symbol 'ω' to refer to the standard natural numbers and to use the symbol ' ' to refer to the first-order part of a possibly nonstandard model of some fragment of arithmetic.We aim to follow this convention.For example, the definitions above use ' ' because they are intended to be interpreted in possibly non-standard models.We use 'ω' when we explicitly build a structure whose first-order part is standard.

REVERSE MATHEMATICS BACKGROUND
We are primarily concerned with the logical relationships among many combinatorial principles provable in the system ACA 0 .Thus we now summarize several of the subsystems of second-order arithmetic below ACA 0 and the relationships among them.

Recursive comprehension, weak König's lemma, and arithmetical comprehension
First we summarize the induction, bounding, and comprehension schemes and three of the most basic subsystems of second-order arithmetic.Everything stated here is explained in full detail in [29].
Full second-order arithmetic consists of the basic axioms: the induction axiom: and the comprehension scheme, which consists of the universal closures of all formulas of the form where ϕ is any formula in the language of second-order arithmetic in which X is not free.We obtain subsystems of second-order arithmetic by limiting induction and comprehension to predicates of a prescribed complexity.
For each n ∈ ω, the Σ 0 n (Π 0 n ) induction scheme, denoted IΣ 0 n (IΠ 0 n ), consists of the universal closures of all formulas of the form where ϕ is Σ 0 n (Π 0 n ).The induction schemes are closely related to the bounding (also called collection) schemes.For each n ∈ ω, the Σ 0 n (Π 0 n ) bounding scheme, denoted BΣ 0 n (BΠ 0 n ), consists of the universal closures of all formulas of the form where ϕ is Σ 0 n (Π 0 n ).The arithmetical comprehension scheme consists of the universal closures of all formulas of the form where ϕ is an arithmetical formula in which X is not free.A further restriction of comprehension is the ∆ 0 1 comprehension scheme, which consists of the universal closures of all formulas of the form where ϕ is Σ 0 1 , ψ is Π 0 1 , and X is not free in ϕ.
RCA 0 (for recursive comprehension axiom) encapsulates recursive mathematics and is the usual base system used when comparing the logical strengths of statements of second-order arithmetic.
The axioms of RCA 0 are the basic axioms, IΣ 0 1 , and the ∆ 0 1 comprehension scheme.
RCA 0 proves sufficient number-theoretic facts to implement the codings of finite sets and sequences that are typical in computability theory.Thus inside RCA 0 , we can fix an enumeration (Φ e ) e∈ of the partial recursive functions.We can also interpret the existence of the set < of all finite strings and give the usual definition of a tree as subset of < that is closed under initial segments.Weak König's lemma (WKL) is the statement "every infinite subtree of 2 < has an infinite path," and WKL 0 is the subsystem RCA 0 + WKL.WKL 0 captures compactness arguments, and it is strictly stronger than RCA 0 (i.e., RCA 0 WKL).
ACA 0 (for arithmetical comprehension axiom) is the subsystem axiomatized by the basic axioms, the induction axiom, and the arithmetical comprehension scheme.It can also be obtained by adding the arithmetical comprehension scheme to RCA 0 .ACA 0 is strictly stronger than WKL 0 , and all of the statements that we consider are provable in ACA 0 .

Ramsey's theorem and its consequences
Let S ⊆ and n ∈ .
[S] n denotes the set of n-element subsets of S, typically thought of as coded by the set of strictly increasing n-tuples over S.
RT n k is the statement "for every coloring f : [ ] n → k, there is an infinite f -homogeneous set." is the statement "for every sequence of sets R, there is an R-cohesive set." For every fixed n ∈ ω with n ≥ 3, the statement (∀k ≥ 2) RT n k is equivalent to ACA 0 over RCA 0 .Indeed, the statement RT 3  2 is already equivalent to ACA 0 over RCA 0 (see [29] Theorem III.7.6).Much work was motivated by the desire to characterize the logical strength of RT 2 2 .Among many results, Cholak, Jockusch, and Slaman [4] (with a bug-fix in [24]) showed that RT 2  2 splits into COH and SRT 2  2 over RCA 0 : RCA 0 ⊢ RT CAC is the statement "every infinite partial order has an infinite chain or an infinite antichain."SCAC is the restriction of CAC to stable partial orders.
Hirschfeldt and Shore give a detailed study of CAC and SCAC (and many other principles) in [15].They show that RCA 0 ⊢ CAC ↔ COH ∧ SCAC and that, over RCA 0 , SCAC is strictly weaker than CAC and CAC is strictly weaker than RT 2 Definition 2.4 (The Erdős-Moser theorem).A tournament T = (D, T ) consists of a set D and an irreflexive binary relation on D such that for all x, y ∈ D with x = y, exactly one of T (x, y) and T ( y, x) holds.(Note the convention that a partial order P = (P, ≤ P ) is identified with its underlying set, whereas a tournament T = (D, T ) identified with its relation.) EM is the statement "for every infinite tournament there is an infinite transitive sub-tournament."SEM is the restriction of EM to stable tournaments.
It is easy to see that RCA 0 ⊢ RT 2  2 → EM and that RCA 0 ⊢ SRT 2 2 → SEM.Furthermore, SEM is strictly weaker than EM over RCA 0 .This can be deduced from the fact that RCA 0 ⊢ EM → 2-DNR [25] (see Section 2.4 below for the definition of 2-DNR) and the fact that there is a (nonstandard) model of RCA 0 + SRT 2 2 (and hence of RCA 0 + SEM) that contains only low sets [5] (see [28] for a complete explanation).By work of Bovykin and Weiermann [3] and of Lerman, Solomon, and Towsner [22], EM and SEM are strictly weaker than RT 2 2 over RCA 0 and are independent of CAC and SCAC over RCA 0 .

Weak weak König's lemma and Martin-Löf randomness
Let T ⊆ 2 < be a tree and let q ∈ .The measure of (the set of paths through) T is ≥ q König's lemma (WWKL), introduced by Yu and Simpson [31] as the restriction of WKL to trees of positive measure, is the statement "every subtree of 2 < with positive measure has an infinite path." WWKL is strictly weaker than WKL over RCA 0 [31].It is well-known that, over RCA 0 , WWKL is equivalent to 1-RAN, which is the statement "for every set X , there is a set Y that is Martin-Löf random relative to X " (see [2], for example).
Avigad, Dean, and Rute [2] generalize WWKL to n-WWKL for each n ∈ ω with n ≥ 1. Informally, n-WWKL asserts that if X is a set and T ⊆ 2 < is a tree of positive measure that is recursive in X n−1 , then T has an infinite path.Care must be taken to formalize n-WWKL without implying the existence of X n−1 or of T .For n ∈ ω with n ≥ 1, let e ∈ X n abbreviate the formula The quantifier 'Q' is '∀' if n is even and is '∃' if n is odd.In the case n = 1, the formula is simply is a formula defining a subtree of 2 < and q ∈ , then that the measure of this tree is ≥ q can be expressed by a formula that states that for every n there is a sequence 〈σ 0 , σ 1 , . . ., σ k−1 〉 of distinct strings in 2 n such that k2 −n ≥ q and (∀i < k)ϕ(σ i ).Similarly, that the measure of the tree defined by ϕ is positive can be expressed by a formula that says that there is a rational q > 0 such that the measure of the tree is ≥ q.Definition 2.5.For n ∈ ω with n ≥ 2, n-WWKL is the statement "for every X and e, if Φ X n−1 e is the characteristic function of a subtree of 2 < with positive measure, then this tree has an infinite path."(That is, there is a function Avigad, Dean, and Rute [2] also generalize 1-RAN to n-RAN, which is a formalization of the statement "for every X there is a Y that is n-random relative to X ," for all n ∈ ω with n ≥ 1.They prove that the correspondence between 1-WWKL and 1-RAN also generalizes to all n once BΣ 0 n is added to n-RAN: for every n ∈ ω with n ≥ 1, n-WWKL and n-RAN + BΣ 0 n are equivalent over RCA 0 .Notice that this implies that for every n ∈ ω with n ≥ 1, RCA 0 +n-WWKL ⊢ BΣ 0 n .

Diagonally non-recursive functions
A function f : → is diagonally non-recursive (DNR) if ∀e( f (e) = Φ e (e)) and is diagonally nonrecursive relative to a set X (DNR(X )) if ∀e( f (e) = Φ X e (e)).An important characterization is that a set computes a DNR function if and only if it computes a fixed-point free function, i.e., a function g : → such that ∀e(W g(e) = W e ).Definition 2.6.DNR is the statement "for every X there is a function f such that ∀e( f (e) = Φ X e (e))." It is clear that no DNR function is recursive and therefore that RCA 0 DNR.On the other hand, it is a classical result of Kučera [21] that every Martin-Löf random set computes a DNR function, and its proof readily relativizes and easily formalizes in RCA 0 .Therefore RCA 0 ⊢ WWKL → DNR.By work of Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman [1], DNR is strictly weaker than WWKL over RCA 0 .
As with weak weak König's lemma and Martin-Löf randomness, we can define a hierarchy of principles expressing the existence of diagonally non-recursive functions.For every n ∈ ω with n ≥ 1, we generalize DNR to n-DNR, which is a formalization of the statement "for every X there exists a function that is diagonally non-recursive relative to X n−1 ."Definition 2.7.n-DNR is the statement "for every X there is a function f such that ∀e( f (e) = Φ X (n−1) e (e))".
Of course, the 'Φ X (n−1) e (e)' in the above definition should be interpreted as it is in Section 2.3.
Proof.Let X be given, and, by n-RAN, let Y be n-random relative to X .Define f : → by ∀e( f (e) = the number whose binary expansion is Y ↾ e).We show that f is almost DNR relative to X (n−1) .Consider the sequence ( i ) i∈ defined by e (e) is σ and σ Z)}.
( i ) i∈ is a uniform sequence of strict (in the sense of [2]) Σ 0,X n sets, and ∀i(µ( i ) ≤ 2 −i ) because i contains at most one string of length e for each e > i.Thus ( i ) i∈ is a Σ 0,X n -test.Therefore Y / ∈ i for some i ∈ .Suppose for a moment that f (e) = Φ X (n−1) e (e) for an e > i.This means that Φ X (n−1) e (e) is the number whose binary expansion is Y ↾ e and thus that Y ↾ e ⊆ i , a contradiction.Therefore f is DNR relative to X (n−1) at all e > i.For each e ≤ i, we can effectively find an index m e such that ∀σ∀x(Φ σ e (e).So we may obtain a function that is DNR relative to X (n−1) by changing f (e) to f (m e ) for all e ≤ i.

Ramsey-type weak König's lemma
In [9], Flood introduced the principle Ramsey-type weak König's lemma, a simultaneous weakening of WKL and RT 2 2 .Informally, RWKL states that if T ⊆ 2 < is an infinite tree, then there is an infinite set X that is either a subset of a path through T or disjoint from a path through T (when thinking of the paths through T as characteristic strings of subsets of ).When formalizing RWKL, care must be taken to avoid implying the existence of a path through T and hence implying WKL.
and a set H ⊆ is homogeneous for an infinite tree T ⊆ 2 < if the tree {σ ∈ T : H is homogeneous for σ} is infinite.RWKL is the statement "for every infinite subtree of 2 < , there is an infinite homogeneous set." Remark 2.10.Flood actually named his principle RKL, for Ramsey-type König's lemma.We found it more convenient to refer to this principle as RWKL.Indeed, we study Ramsey-type variations of several principles, and the convention we follow is to add an 'R' to a principle's name to denote its Ramsey-type variation (see, for example, RSAT, RCOLOR n , and RWWKL below).The typical scheme is to view a combinatorial principle as a problem comprised of instances and solutions to these instances.For example, with WKL, an instance would be an infinite subtree of 2 < , and a solution to that instance would be a path through the tree.The Ramsey-type variation of a principle has the same class of instances, but instead of asking for a full solution in the problem's original sense, we ask only for an infinite set consistent with being a solution.
Flood [9] proved that RCA 0 ⊢ WKL → RWKL and that RCA 0 ⊢ SRT Proof.Let T ⊆ 2 < be an infinite tree.For each s ∈ , let σ s be the leftmost element of T s .We define a tournament R from the tree T .For x < s, if σ s (x) = 1, then R(x, s) holds and R(s, x) fails; otherwise, if σ s (x) = 0, then R(x, s) fails and R(s, x) holds.This tournament R is essentially the same as the coloring f (x, s) = σ s (x) defined by Flood in his proof that RCA 0 ⊢ SRT 2 2 → RWKL ([9] Theorem 5), where he showed that f is stable.By the same argument, R is stable.
The proof now breaks into two cases.First, suppose that the τ(|τ| − 1) for the τ ∈ U < satisfying (⋆) are unbounded.Then, because (⋆) is a Σ 0 1 property of U, there is an infinite set X consisting of numbers of the form τ(|τ| − 1) for τ ∈ U < satisfying (⋆).As argued above, every x ∈ X satisfies R(s, x) for cofinitely many s.Thus we can thin out X to an infinite set H such that (∀x, y ∈ H)(x < y → R( y, x)).Thus H is homogeneous for T with color 0 because H is homogeneous for σ y with color 0 for every y ∈ H.
Flood also proved that RCA 0 ⊢ RWKL → DNR, and this result prompted him to ask if RCA 0 ⊢ DNR → RWKL.Corollary 6.12 shows that the answer to this question is negative.

RAMSEY-TYPE KÖNIG'S LEMMA AND ITS VARIANTS
We investigate the strengths of several variations of RWKL.Our variations are obtained in one of two ways.First, we consider Ramsey-type König's lemma principles applied to different classes of trees.We show that when we restrict to trees of positive measure, the resulting principle is equivalent to DNR (Theorem 3.4); that when we allow subtrees of k < (for a fixed k ∈ ω with k ≥ 2), the resulting principle is equivalent to RWKL (Theorem 3.27); that when we allow bounded subtrees of < , the resulting principle is equivalent to WKL (Theorem 3.19); and that when we allow arbitrary finitely-branching subtrees of < , the resulting principle is equivalent to ACA 0 (Theorem 3.17).Second, we impose additional requirements on the homogeneous sets that RWKL asserts exist.If we require that homogeneous sets be homogeneous for color 0 (and restrict to trees that have no paths that are eventually 1), then the resulting principle is equivalent to WKL (Theorem 3.12).If we impose a bound on the sparsity of the homogeneous sets, then the resulting principle is also equivalent to WKL (Theorem 3.15).If we require that the homogeneous sets be subsets of some prescribed infinite set, then the resulting principle is equivalent to RWKL (Theorem 3.27).It is interesting to note that each variation of RWKL that we consider is either equivalent to RWKL itself or some other well-known statement.We also note that sometimes the Ramsey-type variant of a principle is equivalent to the original principle, as with König's lemma for bounded trees and König's lemma for arbitrary finitely-branching trees; and that sometimes the Ramsey-type variant of a principle is strictly weaker than the original principle, as with weak König's lemma and weak weak König's lemma.
A few of the results in this section begin to hint at robustness in RWKL.For example, we may generalize RWKL to subtrees of k < (for fixed k ∈ ω with k ≥ 2) without changing the principle's strength.We explore the robustness of RWKL more fully in Section 4 and Section 5.This robustness we take as evidence that RWKL is a natural principle.

DNR functions and subsets of paths through trees of positive measure
Just as WKL can be weakened to WWKL by restricting to trees of positive measure, so can RWKL be weakened to RWWKL by restricting to trees of positive measure.Definition 3.1.RWWKL is the statement "for every subtree of 2 < with positive measure, there is an infinite homogeneous set." Applying RWWKL to a tree in which every path is Martin-Löf random yields an infinite subset of a Martin-Löf random set, and every infinite subset of every Martin-Löf random set computes a DNR function.In fact, computing an infinite subset of a Martin-Löf random set is equivalent to computing a DNR function, as the following theorem states.[20], Greenberg and Miller [13]).For every A ∈ 2 ω , A computes a DNR function if and only if A computes an infinite subset of a Martin-Löf random set.Theorem 3.2 also relativizes: a set A computes a DNR(X ) function if and only if it computes an infinite subset of a set that is Martin-Löf random relative to X .Thus one reasonably expects that DNR and RWWKL are equivalent over RCA 0 .This is indeed the case, as we show.The proof makes use of the following recursion-theoretic lemma, which reflects a classical fact concerning diagonally non-recursive functions.

Theorem 3.2 (Kjos-Hanssen
Lemma 3.3.The statement "for every set X there is a function g Notice that in the statement of the above lemma, W X e need not exist as a set.Thus '|W X e | < k' should be interpreted as '∀s(|W X e,s | < k),' where (W X e,s ) s∈ is the standard enumeration of W X e .
. Indeed, Flood's proof uses the construction of a tree of positive measure due to Jockusch [19].(For a similar construction proving a generalization of RWWKL → DNR, see the proof of Lemma 3.6 below.)The proof that DNR → RWWKL is similar to the original proof of Theorem 3.2.However, some adjustments are needed as the original argument uses techniques from measure theory and algorithmic randomness which can only be formalized within WWKL.
We instead use explicit combinatorial bounds.
Assume DNR, and consider a tree T of measure ≥ 2 −c for some c, which we can assume to be ≥ 3 (the reason for this assumption will become clear).For a given set H ⊆ and a value v ∈ {0, 1}, For a tree T and a constant c, let Bad(n, T, c) be the Σ 0 1 predicate 'µ(T ∩Γ 0 n ) < 2 −2c .'In the following claim, {n : Bad(n, T, c)} need not a priori exist as a set, so '|{n : Bad(n, T, c)}| < 2c' should be interpreted in the same manner as Proof.Suppose for a contradiction that |{n : Bad(n, T, c)}| ≥ 2c, and let B be the first 2c elements enumerated in {n : Bad(n, T, c)}.For each n ∈ B, the tree 2 Obtained independently by Flood and Towsner [12].
On the other hand, Putting the two together, we get that 2 Let g be as in Lemma 3.3 for X = T .Given a (canonical index for a) finite set F and a c, we can effectively produce an index e(F, c) such that ∀n(n ∈ W T e(F,c) ↔ Bad(n, T ∩ Γ 0 F , c)).Recursively construct an increasing sequence h 0 < h 1 < h 2 < . . . of numbers by letting, for each s ∈ , H s = and therefore ¬Bad( We show that H is homogeneous for T .Suppose for a contradiction that H is not homogeneous for T .This means that there are only finitely many σ ∈ T such that H is homogeneous for σ.Therefore at some level s, {σ ∈ T s : Fix n ∈ ω with n ≥ 2. Just as with n-WWKL, it is possible to define n-RWWKL to be the generalization of RWWKL to X n−1 -computable trees.The equivalence between n-DNR and n-RWWKL persists in the presence of sufficient induction.
Definition 3.5.For n ∈ ω with n ≥ 2, n-RWWKL is the statement "for every X and e, if is the characteristic function of a subtree of 2 < with positive measure, then there is an infinite homogeneous set."(That is, there is an infinite H ⊆ that is homogeneous for infinitely many Proof.Fix a sequence of functions (b k ) k∈ such that, for each k ∈ , b k is a bijection between and [k] .Let X be given.Let e be an index such that and is the characteristic function of a tree.We need to show that this tree has positive measure.Fix s ∈ .By bounded (σ) = 1} exists as a finite set.For each i < s, the proportion of strings in 2 s missing from T s on account of , so the tree indeed has positive measure.
By n-RWWKL, there is an infinite homogeneous set H for the tree described by Φ X n−1 e .For each i ∈ , let H i denote the set consisting of the i least elements of H. Define f : → by We finish the proof by showing that f is DNR relative to X (n−1) .Suppose for a contradiction that there is an i ∈ such that f (i) = Φ X (n−1) i (i), and let s be such that Φ By the definition of f , we have that b i,s (i).By applying the bijection Proof sketch.Follow the proof that RCA 0 ⊢ DNR → RWWKL from Theorem 3.4, but interpret T as an X (n−1) -computable tree of positive measure in the sense of Section 2.3.The proof of Lemma 3.3 goes through in RCA 0 when X is replaced by X (n−1) and DNR is replaced by n-DNR.The predicate Bad(n, T, c) is now Σ 0 n , and the proof of the claim goes through in RCA 0 + BΣ 0 n .The function g exists by the generalization of Lemma 3.3, and the function e is the same as it was before.The set H is constructed from g and e as it was before.Use IΠ 0 n , a consequence of RCA 0 + IΣ 0 n , to prove the analog of ∀s(µ The rest of the proof is the same as it was before.
Proof.The theorem follows from Lemma 3.6 and Lemma 3.7.
We leave open the question of the exact amount of induction required to prove Lemma 3.6 and Lemma 3.7.It would be particularly interesting to determine whether or not n-RWWKL implies BΣ 0 2 .

Changing homogeneity constraints
Notice that the homogeneous set constructed in the proof of Theorem 3.4 is always homogeneous for color 0, and we could just as easily constructed a set homogeneous for color 1.Thus no additional power is gleaned from RWWKL by prescribing the color of the homogeneous set ahead of time.(i) DNR (ii) RWWKL (iii) For every tree T ⊆ 2 < of positive measure, there is an infinite set that is homogeneous for T with color 0.
One then wonders if any additional strength is gained by modifying RWKL to require that homogeneous sets be homogeneous for color 0. Of course an infinite homogeneous set for color 0 need not exist in general, so we restrict to trees that do not have paths that are eventually 1.For the purposes of the next theorem, "T has no path that is eventually 1" means ∀σ∃n(σ ⌢ 1 n / ∈ T ).
Theorem 3.12.The following statements are equivalent over RCA 0 : (i) WKL (ii) For every infinite tree T ⊆ 2 < with no path that is eventually 1, there is an infinite set homogeneous for T with color 0.
Proof.Clearly (i) → (ii).For (ii) → (i), let S ⊆ 2 < be an infinite tree.We define a tree T ⊆ 2 < whose paths have 0's only at positions corresponding to codes of initial segments of paths through S. Let (τ i ) i∈ be the enumeration of 2 < in length-lexicographic order, and note that ∀i T is a tree because if σ ∈ T is witnessed by τ ∈ S and n < |σ|, then τ ↾ n ∈ S witnesses that σ ↾ n ∈ T .Every string of length n in S witnesses the existence of a string of length n in T , so T is infinite because S is infinite.
We show that T has no path that is eventually 1.Consider a σ ∈ 2 < .Choose m and n such that ∀i Thus T has no path that is eventually 1.By (ii), let H be infinite and homogeneous for T with color 0. If i and j are in H with i ≤ j, then τ i and τ j are in S with τ i τ j .This can be seen by considering a σ ∈ T of length j + 1 for which H is homogeneous with color 0 and a τ ∈ S witnessing that σ ∈ T .Thus we can define an f ∈ 2 by f = i∈H τ i , and this f is a path through S because τ i ∈ S for every i ∈ H.
In [10], Flood studies the computability-theoretic content of Erdős and Galvin's [8] packed versions of Ramsey's theorem.These theorems weaken homogeneity to a property called semihomogeneity, but they require that these semi-homogeneous sets satisfy a certain density requirement.Flood shows that the packed versions of Ramsey's theorem behave similarly to Ramsey's theorem.We formulate a packed version of RWKL and prove that it is equivalent to WKL.For this formulation, we consider an alternate definition of homogeneity.

Definition 3.13. A partial function
If T is an infinite, finitely branching tree, a partial function h: ⊆ → is homogeneous for T if the tree {σ ∈ T : h is homogeneous for σ} is infinite.
In Definition 3.13, we always assume that dom(h) exists as a set.This is no real restriction because in RCA 0 one can prove that every infinite Σ 0 1 -definable set has an infinite subset that actually exists as a set.Thus if h is infinite, we may always restrict h to an infinite subset of dom(h) that exists as a set.If h infinite and homogenous for an infinite tree T ⊆ 2 < , then both of the sets h −1 (0) and h −1 (1) are homogeneous for T , and one of them must be infinite.Conversely, if H is homogeneous for an infinite tree T ⊆ 2 < with color c, then the function h: H → 2 with constant value c is homogeneous for T .Thus, over RCA 0 , it is equivalent to define RWKL in terms of sethomogeneity or in terms of function-homogeneity.However, function-homogeneity lets us impose the density constraints we need for a packed version of RWKL.Function-homogeneity also lets us formulate Ramsey-type variants of full König's lemma and of bounded König's lemma.
Recall that an order function is a non-decreasing unbounded function g : → .

Definition 3.14. Let g be an order function. A partial function
Our packed version of RWKL is equivalent to WKL by an argument that replaces a tree with a version of that tree having sufficient redundancy.
Theorem 3.15.The following statements are equivalent over RCA 0 : (i) WKL (ii) For every order function g satisfying ∀n(g(n) ≤ n) and for every infinite tree T ⊆ 2 < , there is an infinite h that is homogeneous for T and packed for g.Proof.The direction (i) → (ii) is trivial.If f is a path through T , then f is also homogeneous for T and packed for g.
Consider the direction (ii) → (i).Let T be an infinite subtree of 2 < , and let g be an order function bounded by the identity.Define a sequence (u n ) n∈ by u 0 = 0 and The idea behind S is to ensure enough redundancy in it so that the domain of every infinite function that is homogeneous for S and packed for g intersects each interval It is easy to see that if T is infinite, then so is S. To see that S is a tree, consider a σ ∈ S, and let τ ∈ T witness σ's membership in S. Given an n ≤ |σ|, let i = µi(n < u i ) and verify that τ ↾ i witnesses that σ ↾ n is in S. Let h be an infinite function that is homogeneous for S and packed for ) because h is packed for g.By definition, g(u j+1 ) ≥ u j + 1.Thus, by the finite pigeonhole principle, there must be an i in dom(h) ↾ u j+1 with i ≥ u j .Now, for each j, let i j be the least element of dom(h) ∩ [u j , u j+1 ).Define a function f by f ( j) = h(i j ).This f is a path through T .To see this, fix n and let σ ∈ S u n be such that h is homogeneous for σ.Let τ ∈ T witness that σ ∈ S, and note that |τ| = n + 1.For each j < n, we have that σ(i j ) = τ( j) by the choice of i j and the definition of S, and we also have that σ(i j ) = f ( j) by the choice of σ and the definition of f .Thus f ↾ n = τ ↾ n, so f ↾ n ∈ T as desired.

Ramsey-type König's lemma for arbitrary finitely branching trees
Using the functional notion of homogeneity, we easily generalize RWKL to infinite, bounded trees and to infinite, finitely branching trees.It is well known that König's lemma (KL) is equivalent to ACA 0 (see [29] Theorem III.7.2) and that bounded König's lemma (i.e., König's lemma for infinite bounded subtrees of < ) is equivalent to WKL (see [29] Lemma IV.1.4).Interestingly, we find that the Ramsey-type version of König's lemma is equivalent to ACA 0 and that the Ramsey-type version of bounded König's lemma is equivalent to WKL, not RWKL.Definition 3.16.RKL is the statement "for every infinite, finitely branching subtree of < , there is an infinite homogeneous partial function."3Theorem 3.17.RCA 0 ⊢ ACA 0 ↔ RKL.
Proof.We take advantage of the fact that ACA 0 is equivalent to KL over RCA 0 .Clearly RCA 0 ⊢ KL → RKL, so it suffices to show that RCA 0 ⊢ RKL → KL.Thus let T ⊆ < be an infinite, finitely branching tree.Let (τ i ) i∈ be a one-to-one enumeration of < .Define the tree S by Clearly S is a tree.S is infinite because T is infinite and, given a τ in T , it is easy to produce a σ in S of the same length.To see that S is finitely branching, consider a σ ∈ S. For σ ⌢ n to be in S, it must be that τ n is an immediate successor of σ(|σ|−1) on T (or that τ n = in the case that σ = ).
As T is finitely branching and the enumeration (τ n ) n∈ is one-to-one, there are only finitely many such n.
Let h be infinite and homogeneous for S. If i and j are in dom(h) with i ≤ j, then τ h(i) and τ h( j) are in T with τ h(i) τ h( j) , which may be seen by considering a σ ∈ S of length j + 1 such that In fact, ACA 0 is equivalent to KL restricted to trees in which each node has at most two immediate successors (see [29] Theorem III.7.2).Notice that if each node in T has at most two immediate successors, then so does the tree S constructed in the proof of Theorem 3.17.Thus the restriction of RKL to trees in which each node has at most two immediate successors remains equivalent to ACA 0 over RCA 0 .
Recall that a tree T ⊆ < is bounded if there is a function g : → such that (∀σ ∈ T )(∀n < |σ|)(σ(n) < g(n)).Definition 3.18.RbWKL is the statement "for every infinite, bounded subtree of < , there is an infinite homogeneous partial function."Theorem 3.19.RCA 0 ⊢ WKL ↔ RbWKL. 4roof.Over RCA 0 , WKL implies RbWKL because WKL implies bounded König's lemma, which clearly implies RbWKL.Thus it suffices to show that RbWKL implies WKL over RCA 0 .This can be done by following the proof of Theorem 3.17.Let T ⊆ 2 < be an infinite tree.Let (τ i ) i∈ be the enumeration of 2 < in length-lexicographic order, and let g : → be a function such that ∀n, i( ).Thus S is bounded by g.The rest of the proof is identical to that of Theorem 3.17.
We remark that it is not difficult to strengthen Theorem 3.19 by fixing the function bounding the tree in the Ramsey-type bounded König's lemma instance to be an arbitrarily slow growing order function.Indeed, RCA 0 proves the statement "for every order function g, WKL if and only if Ramsey-type König's lemma holds for infinite subtrees of < bounded by g."However, as we will see next, it is not possible to replace an order function by a constant function.

Locality and k-branching trees
We analyze a notion of locality together with Ramsey-type weak König's lemma for k-branching trees.These notions aid our analysis of Ramsey-type analogs of other combinatorial principles.
Consider a function f : [ ] n → k.RT n k asserts the existence of an infinite homogeneous set H ⊆ .However, for the purpose of some application, we may want the infinite homogeneous set H to be a subset of some pre-specified infinite set X ⊆ .This is the idea behind locality, and in such a situation we say that the RT n k -instance f has been localized to X .It is easy to see that RT n k proves that every RT n k -instance can be localized to every infinite X ⊆ .The following proposition is well-known and is often used implicitly, such as when proving RT 2 3 from RT 2 2 .
Proposition 3.20.The following statements are equivalent over RCA 0 : → k and every infinite X ⊆ , there is an infinite H ⊆ X that is homogeneous for f .
Proof.Clearly (ii) → (i), so it suffices to show that (i) → (ii).Let f and X be as in (ii).Let (x i ) i∈ enumerate X in increasing order.Define g : [ ] n → k by g(i 0 , i 1 , . . ., i n−1 ) = f (x i 0 , x i 1 , . . ., x i n−1 ) for increasing n-tuples (i 0 , i 1 , . . ., i n−1 ).Apply RT n k to g to get an infinite H 0 ⊆ that is homogeneous for g with some color c < k.
By analogy with Proposition 3.20, we formulate LRWKL, a localized version of Ramsey-type weak König's lemma.Definition 3.21.LRWKL is the statement "for every infinite tree T ⊆ 2 < and every infinite X ⊆ , there is an infinite H ⊆ X that is homogeneous for T ."Lemma 3.22.RCA 0 ⊢ RWKL ↔ LRWKL.
Let T ⊆ 2 < be an infinite tree and X ⊆ be an infinite set.Let (x i ) i∈ enumerate X in increasing order.Let S ⊆ 2 < be the set S exists by ∆ 0 1 comprehension, and S is clearly closed under initial segments.To see that S is infinite, let n ∈ and, as T is infinite, let τ ∈ T have length x n .Then the σ ∈ 2 n such that we show that H is homogeneous for T with color c.Given n ∈ , let m ∈ H 0 be such that x m > n.By the homogeneity of H 0 for S, let σ ∈ S be of length m and such that (∀i By the definition of S, there is a τ ∈ T of length x m such that (∀i Thus H is homogeneous for τ with color c, and, as |τ| = x m > n, τ ↾ n is a string in T of length n for which H is homogeneous with color c.
Similarly, we can define a localized version of Ramsey-type weak weak König's lemma.
Definition 3.23.LRWWKL is the statement "for every tree T ⊆ 2 < of positive measure and every infinite X ⊆ , there is an infinite H ⊆ X that is homogeneous for T ." Theorem 3.24.The following statements are equivalent over RCA 0 : (i) DNR (ii) RWWKL (iii) LRWWKL.
Proof.Theorem 3.4 states that (i) ↔ (ii), and (iii) → (ii) is clear.To see that (ii) → (iii), we need only check that the the tree S constructed in Lemma 3.22 has positive measure when the tree T has positive measure.To this end, notice that for every k ∈ , which implies that S has positive measure if T has positive measure.
Using LRWKL, we prove versions of RWKL and LRWKL for k-branching trees.Define a set H ⊆ to be homogeneous for a string σ ∈ k < with color c < k and a set H ⊆ to be homogeneous for an infinite tree T ⊆ k < as in Definition 2.9 but with k in place of 2.

Definition 3.25.
− RWKL k is the statement "for every infinite tree T ⊆ k < , there is an infinite H ⊆ that is homogeneous for T ." − LRWKL k is the statement "for every infinite tree T ⊆ k < and every infinite X ⊆ , there is an infinite H ⊆ X that is homogeneous for T ." Lemma 3.26.For every k ∈ ω, RCA 0 ⊢ LRWKL → RWKL k .
Proof.If j < k then RCA 0 ⊢ RWKL k → RWKL j by identifying j < with the obvious subtree of k < .It therefore suffices to show that, for every k ∈ ω, RCA 0 ⊢ RWKL → RWKL 2 k .
Let T ⊆ (2 k ) < be an infinite tree.The idea of the proof is to code T as a subtree of 2 < by coding each number less than 2 k by its binary expansion.We then obtain a homogeneous set for T by using k applications of LRWKL.
For each a < 2 k and each i < k, let a(i) < 2 denote the (i + 1) th digit in the binary expansion of a. Then to each σ ∈ (2 k ) < associate a string τ σ ∈ 2 < of length k|σ| by τ σ (ki + j) = σ(i)( j) (i.e., the j th digit in the binary expansion of σ(i)) for all i < |σ| and all j < k.We define infinite trees and, for each i < k, we find an infinite set H i homogeneous for S i .
Moreover, the sets H i will be such that That is, S 0 consists of the substrings of the binary expansions of the strings in T .S 0 exists by ∆ 0 1 comprehension, S 0 is clearly a tree, and S 0 is infinite because if n ∈ and σ ∈ T has length n, then τ σ ↾ n is a member of S 0 of length n.Let X 0 = {n ∈ : n ≡ 0 mod k}.Apply LRWKL to S 0 and X 0 to get an infinite set H 0 ⊆ X 0 and a color c 0 < 2 such that H 0 is homogeneous for S 0 with color c 0 .Now suppose that S ℓ , H ℓ , and c ℓ are defined for some We show that H is homogeneous for T with color a.
Given n ∈ , let τ ∈ S k−1 be of length kn and such that H k−1 is homogeneous for τ.Let σ ∈ (2 k ) < be such that τ = τ σ .As τ ∈ S k−1 ⊆ S 0 , it must be that σ ∈ T by the definition of S 0 .It remains , and S ℓ+1 was chosen so that if η ∈ S ℓ+1 and m < |η| is in Thus we have the following equivalences.
The statement ∀k RWKL k easily implies RT 1 over RCA 0 , and RT 1 is equivalent to BΣ 0 2 over RCA 0 (this equivalence is due to Hirst [17]).To see that RCA 0 ⊢ ∀k RWKL k → RT 1 , given a The strength of having various kinds homogeneous sets for various kinds of infinite trees is summarized in Table 1.The columns correspond to the kinds of trees allowed, whereas the rows correspond to the kinds of homogeneous sets asserted to exist.The first column considers infinite, finitely branching trees.The second column restricts to trees whose nodes have at most two immediate successors.The third column restricts to trees whose branching is bounded by some function.
The fourth column restricts to trees whose branching is bounded by a constant function.The last column restricts to binary trees of positive measure.The first row corresponds to König-like statements, that is, statements asserting the existence of paths through the tree.The second row asserts the existence of packed homogeneous sets.The third row asserts the existence of sets that are homogeneous for a fixed color.The fourth row asserts the existence of homogeneous sets that are contained in a prescribed infinite set.The last row asserts the existence of homogeneous sets.One can conceivably consider a Ramsey-type variant of any Π 1 2 statement ∀X ∃Y ϕ(X , Y ) so long as one can provide a reasonable formulation of what it means for a set Z to be consistent with a Y such that ϕ(X , Y ).For example, in the case of RWKL, we think of a set H as being consistent with a path through an infinite tree T ⊆ 2 < if H is homogeneous for T .We are interested in analyzing the strengths of Ramsey-type variants of statements that are equivalent to WKL over RCA 0 .Several such statements have trivial Ramsey-type variants.For example, RCA 0 proves that for every pair of injections f , g : → with disjoint ranges, there is an infinite set X consistent with being a separating set for the ranges of f and g because RCA 0 proves that there is an infinite subset of the range of f .The obvious Ramsey-type variant of Lindenbaum's lemma (every consistent set of sentences has a completion) is also easily seen to be provable in RCA 0 .For the remainder of this paper, we consider non-trivial Ramsey-type variants of the compactness theorem for propositional logic and of graph coloring theorems.

Definition 4.1.
A set C of propositional formulas is finitely satisfiable if every finite C 0 ⊆ C is satisfiable (i.e., has a satisfying truth assignment).We denote by SAT the compactness theorem for propositional logic, which is the statement "every finitely satisfiable set of propositional formulas is satisfiable." It is well-known that SAT is equivalent to WKL over RCA 0 (see [29] Theorem IV.3.3).
If C is a set of propositional formulas, then let atoms(C) denote the set of propositional atoms appearing in the formulas in C. Strictly speaking, RCA 0 does not in prove that atoms(C) exists for every set of propositional formulas C.However, in RCA 0 we can rename the atoms appearing in a set of propositional formulas C in such a way as to produce an equivalent set of propositional formulas C ′ for which atoms(C ′ ) does exist.Indeed, we may assume that atoms(C) = whenever atoms(C) is infinite.Thus for ease of mind we always assume that atoms(C) exists as a set.

Definition 4.2. Let C be a set of propositional formulas. A set H ⊆ atoms(C) is homogeneous for C
if there is a c ∈ {T, F} such that every finite C 0 ⊆ C is satisfiable by a truth assignment ν such that As is typical, we identify T with 1 and F with 0.

Definition 4.3.
− RSAT is the statement "for every finitely satisfiable set C of propositional formulas with atoms(C) infinite, there is an infinite H ⊆ atoms(C) that is homogeneous for C." − LRSAT is the statement "for every finitely satisfiable set C of propositional formulas with atoms(C) infinite and every infinite X ⊆ atoms(C), there is an infinite H ⊆ X that is homogeneous for C." We also consider r.e.versions of RSAT and LRSAT, denoted r.e.-RSAT and r.e.-LRSAT, obtained by replacing the finitely satisfiable set of propositional formulas C by a list of propositional formulas (ϕ i ) i∈ such that {ϕ i : i < n} is satisfiable for every n ∈ .This amounts to considering r.e.sets of propositional formulas instead of recursive sets of propositional formulas.In this situation, we may still assume that atoms((ϕ i ) i∈ ) (the set of propositional atoms appearing in the ϕ i 's) exists as a set.
We first show that RCA 0 ⊢ RSAT → RWKL.In fact, we show that the restriction of RSAT to what we call 2-branching clauses implies RWKL over RCA 0 .This technical restriction is useful for the proof of Theorem 5.13 in our analysis of Ramsey-type graph coloring principles.
Recall that a propositional formula ℓ is called a literal if either ℓ = a or ℓ = ¬a for some propositional atom a and that a clause is a disjunction of literals.Proof.Let A = {a i : i ∈ } be a set of propositional atoms, and to each string σ ∈ 2 < associate the clause θ σ = i<|σ| ℓ i , where ∈ T }, and observe that C is 2-branching.We show that C is finitely satisfiable.Given C 0 ⊆ C finite, choose n large enough so that the atoms appearing in the clauses in C 0 are among {a i : i < n}.As T is infinite, choose a τ ∈ T of length n.Define a truth assignment t : {a i : i < n} → {T, F} by t(a i ) = τ(i).Now, if θ is a clause in C 0 , then θ = θ σ = i<|σ| ℓ i for some σ / ∈ T with |σ| < n.Thus there is an i < n such that σ(i) = τ(i) (because τ ∈ T and σ / ∈ T ), from which we see that t(ℓ i ) = T and hence that t(θ σ ) = T. Thus t satisfies C 0 .By RSAT 2-branching , let H 0 ⊆ A and c ∈ {T, F} be such that H 0 is homogeneous for C with truth value c.Let H = {i ∈ : a i ∈ H 0 }.We show that H is homogeneous for a path through T with color c.Given n ∈ , we want to find a τ ∈ T such that |τ| = n and Thus let t : {a i : i < n} → {T, F} be a truth assignment satisfying Thus τ ∈ T as desired.
Proof.Let (ϕ i ) i∈ be a list of propositional formulas over an infinite set of atoms A such that {ϕ i : i < n} is satisfiable for every n ∈ , and let X ⊆ A be infinite.Let (a i ) i∈ enumerate A. For each σ ∈ 2 < , identify σ with the truth assignment ν σ on {a i : i < |σ|} given by (∀i < |σ|)(ν σ (a i ) = T ↔ σ(i) = 1).Let T ⊆ 2 < be the tree where ν σ (ϕ i ) is the truth value assigned to ϕ i by ν σ (we consider ν σ (ϕ i ) undefined-hence not Fif ϕ i contains an atom a m for an m ≥ |σ|).T exists by ∆ 0 1 comprehension and is downward closed.
T is infinite because for any n ∈ , any satisfying truth assignment of {ϕ i : i < n} restricted to {a i : i < n} yields a string in T of length n.Let X 0 = {i ∈ : a i ∈ X }, and, by LRWKL, let H 0 ⊆ X 0 and c < 2 be such that H 0 is infinite and homogeneous for T with color c.Let H = {a i : i ∈ H 0 } and note that it is an infinite subset of X .We show that, for every n ∈ , {ϕ i : i < n} can be satisfied by a truth assignment ν such that (∀a ∈ H)(ν(a) = c).Let n ∈ , and let m be large enough so that atoms({ϕ i : i < n}) ⊆ {a i : i < m}.Let σ ∈ T be such that |σ| = m and H 0 is homogeneous for Proof.Clearly (v) → (iii) → (ii) and (v) → (iv) → (ii), so it suffices to show the equivalence of (i), (ii), and (v).We have that (i) → (v) by Proposition 4.6 and Lemma 3.22, that (v) → (ii) is clear, and that (ii) → (i) by Proposition 4.5.

RAMSEY-TYPE GRAPH COLORING PRINCIPLES
Let k ∈ , and let G = (V, E) be a graph.A function f : , and a graph is locally k-colorable if every finite subgraph is k-colorable.A simple compactness argument proves that every locally k-colorable graph is k-colorable.In the context of reverse mathematics, we have the following well-known equivalence.
Theorem 5.1 (see [18]).For every k ∈ ω with k ≥ 2, the following statements are equivalent over In light of Theorem 5.1, we define Ramsey-type analogs of graph coloring principles and compare them to Ramsey-type weak König's lemma.
subgraph that is k-colorable by a coloring that colors every vertex in V 0 ∩ H color 0. We often write homogeneous for k-homogeneous when the k is clear from context.
− RCOLOR k is the statement "for every infinite, locally k-colorable graph G = (V, E), there is − LRCOLOR k is the statement "for every infinite, locally k-colorable graph G = (V, E) and every infinite X ⊆ V , there is an infinite H ⊆ X that is k-homogeneous for G." The goal of this section is to obtain the analog of Theorem 5.1 with RWKL in place of WKL and with RCOLOR k in place of the statement "every locally k-colorable graph is k-colorable."We are able to obtain this analog for all standard k ≥ 3 instead of all standard k ≥ 2. The case k = 2 remains open.Showing the forward direction, that RCA 0 ⊢ RWKL → RCOLOR k (indeed, that RCA 0 ⊢ RWKL → LRCOLOR k ), is straightforward.
Proof.Let G = (V, E) be an infinite graph such that every finite V 0 ⊆ V induces a k-colorable subgraph, and let X ⊆ V be infinite.Enumerate V as (v i ) i∈ , and let T ⊆ k < be the tree T exists by ∆ 0 1 comprehension and is downward closed.T is infinite because for any n ∈ , any k-coloring of the subgraph induced by {v i : i < n} corresponds to a string in the tree of length n.Let X 0 = {i ∈ : v i ∈ X }, and apply LRWKL k (which follows from RCA 0 + RWKL by Theorem 3.27) to T and X 0 to get an infinite set H 0 ⊆ X 0 and a color c < k such that H 0 is homogeneous for a path through T with color c.Let H = {v i : i ∈ H 0 }.We show that every finite V 0 ⊆ V induces a subgraph that is k-colorable by a coloring that colors every v ∈ V 0 ∩ H color 0. Let V 0 ⊆ V be finite, let n = max{i + 1 : v i ∈ V 0 }, and let σ ∈ T be such that |σ| = n and such that H 0 is homogeneous for σ with color c.Then the coloring of V 0 given by v i → σ(i) is a k-coloring of V 0 that colors the elements of V 0 ∩ H color c.Swapping colors 0 and c thus gives a k-coloring of V 0 that colors the elements of V 0 ∩ H color 0.
We now prove that RCA 0 ⊢ RCOLOR 3 → RWKL (Theorem 5.13 below).Our proof factors through the Ramsey-type satisfiability principles and is a rather elaborate exercise in circuit design.
The plan is to prove that RCA 0 ⊢ RCOLOR 3 → RSAT 2-branching , then appeal to Proposition 4.5.
Given a 2-branching set of clauses C, we compute a locally 3-colorable graph G such that every set homogeneous for G computes a set that is homogeneous for C. G is built by connecting widgets, which are finite graphs whose colorings have desirous properties.A widget W ( v) has distinguished vertices v through which we connect the widget to the larger graph.These distinguished vertices can also be regarded, in a sense, as the inputs and outputs of the widget.
In an RCOLOR 3 instance built out of widgets according to an RSAT 2-branching instance, some of the vertices code literals so that the colorings of these coding vertices code truth assignments of the corresponding literals in such a way that a homogeneous set for the RSAT 2-branching instance can be decoded from a homogeneous set for the graph that contains only coding vertices.However, we have no control over what vertices appear in an arbitrary homogeneous set.Therefore, we must build our graph so that the color of every vertex gives information about the color of some coding vertex.
When we introduce a widget, we prove a lemma concerning the three key aspects of the widget's operation: soundness, completeness, and reversibility.By soundness, we mean conditions on the 3-colorings of the widget, which we think of as input-output requirements for the widget.By completeness, we mean that the widget is indeed 3-colorable and, moreover, that 3-colorings of certain sub-widgets extend to 3-colorings of the whole widget.By reversibility, we mean that the colors of some vertices may be deduced from the colors of other vertices.
To aid the analysis of our widgets, we introduce a notation for the property that a coloring colors two vertices the same color.The graph G that we build from widgets has three distinguished vertices 0, 1, and 2, connected as a triangle.The intention of these vertices is to code truth values.If v is a vertex coding a literal ℓ, then (v, 2) is an edge in G, and, for a 3-coloring ν, we interpret v = ν 0 as ℓ is false and v = ν 1 as ℓ is true.Our widgets often include vertices 0, 1, and 2. Proof.The lemma follows from examining the two possible (up to permutations of the colors) 3colorings of Rx → y y →z (a, u): We see (i) immediately.For (ii), if a = ν x, then color the widget according to the first coloring; and if a = ν y, then color the widget according to the second coloring.For (iii), The intention is that, in Rx → y y →z (a, u), the vertices x, y, and z are some permutation of the vertices 0, 1, and 2. For example, R 0 →1 1 →2 (a, u) is the instance of this widget where x = 0, y = 1, and z = 2.
The notation 'R 0 →1 1 →2 (a, u)' is evocative of Lemma 5.6 (i).Thinking of a as the widget's input and of u as the widget's output, Lemma 5.6 (i) says that the widget maps 0 to 1 and maps 1 to 2.
Widget 5.7.U x, y,z (ℓ, b, u) is the following widget.(ℓ, r), and some of the edges incident to them (for example, the edge (x, r)) have been omitted to improve legibility.
The properties of U x, y,z (ℓ, b, u) highlighted by the next lemmas may seem ill-motivated at first.
We explain their significance after the proofs.In each of the above cases, the sub-widget Rx → y y →z (ℓ, r) is colored according to Lemma 5.6.
For (ii), let ν be a 3-coloring of U x, y,z (ℓ, b, u) in which ℓ = ν x and b = ν y.Then it must be that l = ν y and d = ν z, and therefore it must be that u = ν x.
Item (iii) can be seen by inspecting the first and third colorings in the proof of (i).
Item (iv) can be seen by inspecting the last three colorings in the proof of (i).
Lemma 5.9.Let ν be a 3-coloring of U x, y,z (ℓ, b, u).If w is l, u, or any vertex appearing in the Rx → y y →z (ℓ, r) sub-widget that is not x, y, or z, then the color of w determines the color of ℓ.Moreover, (ℓ, r) that is not x, y, or z, then the color of w determines the color of ℓ by Lemma 5.6 (iii).For u, if u = ν x or u = ν z it cannot be that ℓ = ν y because then l = ν x and, by Lemma 5.6 (i), r = ν z.On the other hand, if u = ν y, it cannot be that ℓ = ν x because then l = ν y.
The idea is to code truth assignments that satisfy the clause as 3-colorings of a graph constructed by chaining together widgets of the form U x, y,z (ℓ i , b, u).Let ν be a 3-coloring of U x, y,z (ℓ i , b, u).The color of the vertex ℓ i represents the truth value of the literal ℓ i : ℓ i = ν x is interpreted as ℓ i is false, and ℓ i = ν y is interpreted as ℓ i is true.The color of the vertex b represents the truth value of ℓ 0 ∨ ℓ 1 ∨ • • • ∨ ℓ i−1 as well as the truth value of the literal false (and hence also as ℓ i−1 is false); and b = ν z is interpreted as ℓ i−1 is true (and hence also as Similarly, the color of the vertex u represents the truth value of ℓ 0 ∨ ℓ 1 ∨ • • • ∨ ℓ i as well as the truth value of the literal ℓ i .However, the meanings of the colors are permuted: u = ν x is interpreted as ℓ 0 ∨ ℓ 1 ∨ • • • ∨ ℓ i is false (and hence also as ℓ i is false); u = ν y is interpreted as ℓ i is true (and hence also as true but ℓ i is false.Lemma 5.8 tells us that U x, y,z (ℓ i , b, u) properly implements this coding scheme.Lemma 5.8 (ii) says that if a 3-coloring codes that ℓ i is false and The reader may worry that here it is also possible to extend ν to incorrectly code that ℓ 0 ∨ ℓ 1 ∨ • • • ∨ ℓ i is false, so we assure the reader that this is irrelevant.What is important is that it is possible to extend ν to code the correct information.Lemma 5.8 (iv) says that if ν is a 3-coloring of the subgraph of U x, y,z (ℓ i , b, u) induced by {x, y, z, ℓ i , b} coding that ℓ i is true, then ν can be extended to a 3-coloring of Lemma 5.9 helps us deduce the colors of literal-coding vertices from the colors of auxiliary vertices and hence helps us compute a homogeneous set for a set of clauses from a homogeneous set for a graph.
The next widget combines U x, y,z (ℓ, b, u) widgets into widgets coding clauses.
The widget also contains the edge (2, ℓ i ) for each i < n, which we omitted from the diagram to keep it legible.For 0 < i < n, the sub-widget , and is 1 if x ≡ 2 mod 3. Note that the vertex x is thus drawn twice because it is identical to one of 0, 1, 2. For clarity, we also point out that in the case of D(ℓ 0 ), the widget is simply 0 1 2 ℓ 0 .

then the color of w determines either the color of ℓ i or the color of ℓ
Proof.Consider a 3-coloring ν of D(ℓ 0 , ℓ 1 , . . ., ℓ n−1 ), an i with 0 < i < n, and a vertex w in an , then the color of w determines the color of ℓ i by Lemma 5.6 (iii).If w appears in U i (ℓ ′ i , u i−1 , u i ), then there are a few cases.If w is not u i−1 or d, then the color of w determines the color ℓ ′ i by Lemma 5.9, which we have just seen determines the color of ℓ i (or ℓ ′ i is ℓ i in the case i ≡ 0 mod 3).Consider w = u i−1 .If i = 1, then u i−1 is really ℓ 0 , and of course the color of ℓ 0 determines the color of ℓ 0 .Otherwise, i > 1, u i−1 appears in the sub-widget , and hence the color of u i−1 determines the color of ℓ i−1 .

Lastly, consider w
, where x, y, and z are some permutation of 0, 1, and 2. If d = ν x or d = ν y, then this determines the color of ℓ ′ i by Lemma 5.9, which in turn determines the color of ℓ i .Otherwise d = ν z, meaning that u i−1 = ν z by Lemma 5.9.If i = 1, then z = 1, u 0 is really ℓ 0 , and we conclude that ) and, by examining the proof of Lemma 5.9, u i−1 = ν z implies that ℓ ′ i−1 = ν y, which in turn determines the color of ℓ i−1 .
Let C be a 2-branching and finitely satisfiable set of clauses over an infinite set of atoms A = {a i : i ∈ }.We assume that no clause in C is a proper prefix of any other clause in C by removing from C every clause that has a proper prefix also in C. We build a locally 3-colorable graph G such that every infinite homogeneous set for G computes an infinite homogeneous set for C. To start, G contains the vertices 0, 1, and 2, as well as the literal-coding vertices a i and ¬a i for each atom a i ∈ A. These vertices are connected according to the diagram below.
Then add the widget D(ℓ 0 , ℓ 1 , . . ., ℓ n−1 ) by overlapping it with D(s 0 , s 1 , . . ., s m−1 ) as described above.In D(ℓ 0 , ℓ 1 , . . ., ℓ n−1 ), for each i < n, the vertex ℓ i is the vertex a i if the literal ℓ i is the literal a i , and the vertex ℓ i is the vertex ¬a i if the literal ℓ i is the literal ¬a i .The vertices appearing in the sub-widgets R i (ℓ i , ℓ ′ i ) and U i (ℓ ′ i , u i−1 , u i ) for i beyond the index at which chosen fresh, except for 0, 1, 2, and the literal-coding vertices ℓ i .This completes the construction of G.

Claim. G is locally 3-colorable.
Proof.Let G 0 be a finite subgraph of G. Let s be the latest stage at which a vertex in G 0 appears, and let C 0 ⊆ C be the set of clauses considered up to stage s.By extending G 0 , we may assume that it is the graph constructed up to stage s.
By the finite satisfiability of C, let t : atoms(C 0 ) → {T, F} be a truth assignment satisfying C 0 .
The hypothesis of Lemma 5.11 (i) is satisfied because t satisfies C 0 , so for each clause ℓ 0 ∨ ℓ 1 ∨ • • • ∨ ℓ n−1 in C 0 , there is an i < n such that ℓ i = ν 1. Overlapping widgets D(ℓ 0 , ℓ 1 , . . ., ℓ n−1 ) and D(s 0 , s 1 , . . ., s m−1 ) are colored consistently because the colors of the shared vertices depend only on the colors of the literal-coding vertices corresponding to the longest common prefix of the two clauses.
Apply RCOLOR 3 to G to get an infinite homogeneous set H. We may assume that H contains exactly one of the truth value-coding vertices 0, 1, or 2. Call this vertex c.
Consider a vertex w ∈ H that is not c.The vertex w appears in some widget D(ℓ 0 , ℓ 1 , . . ., ℓ n−1 ), and, by Lemma 5.12, from w we can compute an i < n and a c i ∈ {0, 1} such that ℓ i = ν c i whenever ν is a 3-coloring of D(ℓ 0 , ℓ 1 , . . ., ℓ n−1 ) in which w = ν c.Moreover, for each literal ℓ, we can compute a bound on the number of vertices w in the graph whose color determines the color of ℓ.Still by Lemma 5.12, if w appears in an R i (ℓ i , ℓ ′ i ) sub-widget or a U i (ℓ ′ i , u i−1 , u i ) sub-widget, then the color of w determines either the color of ℓ i or the color of ℓ i−1 .Thus the vertices whose colors determine the color of ℓ i only appear in ), and U i+1 (ℓ ′ i+1 , u i , u i+1 ) sub-widgets.The fact that C is a 2-branching set of clauses and our protocol for overlapping the D(ℓ 0 , ℓ 1 , . . ., ℓ n−1 ) widgets together imply that, for every j > 0, there are at most 2 j sub-widgets of the form R j (ℓ j , ℓ ′ j ) and at most 2 j sub-widgets of the form U j (ℓ ′ j , u j−1 , u j ).This induces the desired bound on the number of vertices whose colors determine the color of ℓ i .
Thus from H we can compute an infinite set H ′ of pairs 〈ℓ, c ℓ 〉, where each ℓ is a literal-coding vertex and each c ℓ is either 0 or 1, such that every finite subgraph of G is 3-colorable by a coloring ν such that (∀〈ℓ, c ℓ 〉 ∈ H ′ )(ℓ = ν c ℓ ).Modify H ′ to contain only pairs 〈a, c a 〉 for positive literal-coding vertices a by replacing each pair of the form 〈¬a, c ¬a 〉 with 〈a, 1 − c ¬a 〉.Now apply the infinite pigeonhole principle to H ′ to get an infinite set H ′′ of positive literal-coding vertices a and a new c ∈ {0, 1} such that the corresponding c a is always c.We identify a positive literal-coding vertex a with the corresponding atom and show that H ′′ is homogeneous for C.
RCA 0 suffices to prove that a finite graph is 2-colorable if and only if it does not contain an odd-length cycle.Thus the condition that every finite subset of vertices of a graph induces a 2colorable subgraph is equivalent to the condition that the graph does not contain an odd-length cycle.Moreover, if G = (V, E) is a graph such that every finite subset of V induces a 2-colorable subgraph, then, for any H ⊆ V , every finite V 0 ⊆ V induces a subgraph that is 2-colorable by a coloring that colors every v ∈ V 0 ∩ H color 0 if and only if no two elements of H are connected by an odd-length path.Thus, over RCA 0 , we immediately have the following two equivalences:
In summary, the situation is thus.WKL and RT 2 2 each imply RWKL and therefore each imply RCOLOR 2 .However, if WKL is weakened to WWKL, then it no longer implies RCOLOR 2 .Similarly, if RT 2  2 is weakened to CAC, then it no longer implies RCOLOR 2 .We begin our analysis of RCOLOR 2 by constructing an infinite, recursive, bipartite graph with no infinite, recursive, homogeneous set.It follows that RCA 0 RCOLOR 2 .The graph we construct avoids potential infinite, r.e., homogeneous sets in a strong way that aids our proof that RCA 0 + CAC RCOLOR 2 .Definition 6.1.Let G = (V, E) be an infinite graph.A set W ⊆ V 2 is column-wise homogeneous for G if W [x] is infinite for infinitely many x (where W [x] = y : x, y ∈ W is the x th column of W ), and ∀x∀ y( y ∈ W [x] → x, y is homogeneous for G).

Lemma 6.2.
There is an infinite, recursive, bipartite graph G = (ω, E) such that no r.e.set is columnwise homogeneous for G.
Proof.The construction proceeds in stages, starting at stage 0 with E = .We say that W e requires attention at stage s if e < s and there is a least pair x, y such that e,s , − x and y are not connected to each other, and − neither x nor y is connected to a vertex ≤ e.
At stage s, let e be least such that W e requires attention at stage s and has not previously received attention.W e then receives attention by letting x, y witness that W e requires attention at stage s, letting u and v be the least isolated vertices > s, and adding edges (x, u), (u, v), and (v, y) to E. This completes the construction.
We verify the construction.We first show that G is acyclic by showing that it is acyclic at every stage.It follows that G is bipartite because a graph is bipartite if and only if it has no odd cycles.All vertices are isolated at the beginning of stage 0, hence G is acyclic at the beginning of stage 0. By induction, suppose G is acyclic at the beginning of stage s.If no W e requires attention at stage s, then no edges are added at stage s, hence G is acyclic at the beginning of stage s + 1.If some least W e requires attention at stage s, then during stage s we add a length-3 path connecting the connected components of the x and y such that x, y witnesses that W e requires attention at stage s.This action does not add a cycle because by the definition of requiring attention, x and y are not connected at the beginning of stage s.Hence G is acyclic at the beginning of stage s + 1.
We now show that, for every e, if there are infinitely many x such that W [x]   e is infinite, then there are an x and a y with y ∈ W [x]   e and x, y not homogeneous for G.If W e receives attention, then there is a length-3 path between an x and a y with y ∈ W [x]  e , in which case x, y is not homogeneous for G. Thus it suffices to show that if W [x]   e is infinite for infinitely many x, then W e requires attention at some stage.

Suppose that W [x]
e is infinite for infinitely many x, and suppose for a contradiction that W [x]   e never requires attention.Let s 0 be a stage by which every W i for i < e that ever requires attention has received attention.The graph contains only finitely many edges at each stage, so let x 0 be an upper bound for the vertices that are connected to the vertices ≤ e at stage s 0 .Notice that when some W i receives attention, the vertices connected at that stage are not connected to vertices ≤ i.
Therefore once all the W i for i < e that ever require attention have received attention, no vertex that is not connected to a vertex ≤ e is ever connected to a vertex ≤ e.In particular, no vertex ≥ x 0 is ever connected to a vertex ≤ e.Now let x > x 0 be such that W [x]   e is infinite, and let s 1 > s 0 be a stage by which every W i for i < x that ever requires attention has received attention.Let y 0 be an upper bound for the vertices that are connected to x and the vertices ≤ e at stage s 1 , and again note that no vertex ≥ y 0 is ever connected to x or a vertex ≤ e.As W [x]   e is infinite, let s > s 1 be a stage at which there is a y > y 0 with x < y < s and y ∈ W [x]  e,s .This y is not connected to x, and neither x nor y is connected to a vertex ≤ e, so W e requires attention at stage s, a contradiction.Proposition 6.3.RCA 0 RCOLOR 2 .
Proof.Consider the ω-model of RCA 0 whose second-order part consists of exactly the recursive sets.The graph G from Lemma 6.2 is in the model because G is recursive.However, the model contains no homogeneous set for G because if H were an infinite, recursive, homogeneous set, then {〈x, y〉 : x, y ∈ H} would be a recursive, column-wise homogeneous set, thus contradicting Lemma 6.2.
The notion of restricted Π 1 2 conservativity helps separate Ramsey-type weak König's lemma and the Ramsey-type coloring principles from a variety of weak principles.Definition 6.4 (see [15,16]).
We now adapt the proof that RCA 0 + CAC DNR in [15] to prove that RCA 0 + CAC RCOLOR 2 .We build an ω-model of RCA 0 + SCAC + COH that is not a model of RCOLOR 2 by alternating between adding chains or antichains to stable partial orders and adding cohesive sets without ever adding an infinite set homogeneous for the graph from Lemma 6.2.Lemma 6.7.Let X be a set, let G = (V, E) be a graph recursive in X such that no column-wise homogeneous set for G is r.e. in X , and let P = (P, ≤ P ) be an infinite, stable partial order recursive in X .Then there is an infinite C ⊆ P that is either a chain or an antichain such that no column-wise homogeneous set for G is r.e. in X ⊕ C. Proof.For simplicity, assume that X is recursive.The proof relativizes to non-recursive X .As P is stable, assume for the sake of argument that P satisfies (∀i The case with ≥ P in place of ≤ P is symmetric.Also assume that there is no recursive, infinite antichain C ⊆ P, for otherwise we are done. Let U = i ∈ P : (∃s)(∀ j > s)( j ∈ P → i ≤ P j) .The fact that there is no recursive, infinite antichain in P implies that U is infinite.Let F = (F, ⊑) be the partial order consisting of all σ ∈ U <ω that are increasing in both < and ≤ P , where τ ⊑ σ if τ σ.Let H be sufficiently generic for F, and notice that H (or rather, the range of H, which is computable from H as H is increasing in <) is an infinite chain in P. Suppose for a contradiction that W H e is column-wise homogeneous for G. Fix a σ H such that σ ∀x∀ y( y ∈ (W H e ) [x] → x, y is homogeneous for G).
Define a partial computable function τ: ω 2 → P <ω by letting τ(x, i) ∈ P <ω be the string with the least code such that τ(x, i) ⊇ σ, that τ(x, i) is increasing in both < and ≤ P , and that |(W τ(x,i) e ) [x] | > i.From here there are two cases: Case 1: There are infinitely many pairs 〈x, i〉 such that both τ(x, i) is defined and there is a y ∈ (W τ(x,i) e ) [x] with x, y not homogeneous for G.The last element of such a τ(x, i) is in P U because otherwise τ(x, i) ∈ F and τ(x, i) σ, contradicting that σ ∀x∀ y( y ∈ (W H e ) [x] → x, y is homogeneous for G).Thus the set C consisting of the last elements of such strings τ(x, i) is an infinite r.e.subset of P U.As elements i of P U have the property (∃s)(∀ j > s)( j ∈ P → i | P j), we can thin C to an infinite r.e.antichain in P and hence to an infinite recursive antichain in P, a contradiction.
Case 2: There are finitely many pairs 〈x, i〉 such that τ(x, i) is defined and there is a y ∈ (W τ(x,i) e ) [x] with x, y not homogeneous for G.In this case, let x 0 be such that if x > x 0 and τ(x, i) Then W is an r.e.set that is column-wise homogeneous for G, a contradiction.
Thus there is no column-wise homogeneous set for G that is r.e. in H. Therefore (the range of) H is our desired chain C. Lemma 6.8.Let X be a set, let G = (V, E) be a graph recursive in X such that no column-wise homogeneous set for G is r.e. in X , and let R = (R i ) i∈ω be a sequence of sets uniformly recursive in X .
Then there is an infinite set C that is cohesive for R such that no column-wise homogeneous set for G is r.e. in X ⊕ C. Proof.For simplicity, assume that X is recursive.The proof relativizes to non-recursive X .
We force with recursive Mathias conditions (D, L), where D ⊆ ω is finite, L ⊆ ω is infinite and recursive, and every element of D is less than every element of L. The order is (D 1 , L 1 ) ⊑ (D 0 , L 0 ) if D 0 ⊆ D 1 , L 1 ⊆ L 0 , and D 1 D 0 ⊆ L 0 .Let H be sufficiently generic.Then H is an infinite cohesive set for R (as in, for example, Section 4 of [4]).
Suppose for a contradiction that W H e is column-wise homogeneous for G. Let (D, L) be a condition such that D ⊆ H ⊆ L and (D, L) ∀x∀ y( y ∈ (W H e ) [x] → x, y is homogeneous for G).
W is an r.e.set, and ∀x∀ y( y ∈ W [x] → x, y is homogeneous for G).To see the second statement, suppose there is a x, y ∈ W such that x, y is not homogeneous for G, and let E witness x, y ∈ W . Then (E, L E) (D, L), but (E, L E) ( y ∈ (W H e ) [x] ∧ x, y is not homogeneous for G), a contradiction.Finally, W ⊇ W H e because if x, y ∈ W H e , then there is a finite E with D ⊆ E ⊆ L such that x, y ∈ W E e , in which case x, y ∈ W . Thus W is an r.e.set that is column-wise homogeneous for G.This contradicts the lemma's hypothesis.Therefore no column-wise homogeneous set for G is r.e. in H, so H is the desired cohesive set.Theorem 6.9.RCA 0 + CAC RCOLOR 2 Proof.Iterate and dovetail applications of Lemma 6.7 and Lemma 6.8 to build a collection of sets such that (ω, ) RCA 0 + COH + SCAC, the graph G from Lemma 6.2 is in , and no set that is r.e. in any set in is column-wise homogeneous for G. Then (ω, ) CAC by [15], and (ω, ) RCOLOR 2 by the same argument as in Proposition 6.3.
We conclude by proving that RCA 0 + DNR RWKL, thereby answering [9] Question 9.In fact, we prove the stronger result RCA 0 + WWKL RCOLOR 2 .This is accomplished by building a recursive bipartite graph G such that the measure of the set of oracles that compute homogeneous sets for G is 0. It follows that there is a Martin-Löf random X that does not compute a homogenous set for G, and a model of RCA 0 + WWKL +¬ RCOLOR 2 is then easily built from the columns of X .
Recall that, in the context of a bipartite graph G = (V, E), a set H ⊆ V is 2-homogeneous for G if no two vertices in H are connected by an odd-length path in G.Here we simply say that such an H is G-homogeneous (or just homogeneous).Likewise, if H ⊆ V contains two vertices that are connected by an odd-length path in G, then H is G-inhomogeneous (or just inhomogeneous).Theorem 6.10.There is a recursive bipartite graph G = (ω, E) such that the measure of the set of oracles that enumerate homogeneous sets for G is 0.
Proof.By Lebesgue density considerations (see, for example, [26] Theorem 1.9.4), if a positive measure of oracles enumerate infinite homogeneous sets for a graph G, then (∀ε > 0)(∃e)[µ{X : W X e is infinite and G-homogeneous} Thus it suffices to build G to satisfy the following requirement R e for each e ∈ ω: R e : µ{X : W X e is infinite and G-homogeneous} ≤ 0.9.
Let us first give a rough outline of the construction.Observe our construction must necessarily produce a graph G that does not contain an infinite connected component.If G has an infinite connected component, then that component contains a vertex v such that infinitely many vertices are connected to v by an even-length path.These vertices that are at an even distance from v can be effectively enumerated, and they form a homogeneous set.Thus our graph G must be a union of countably many finite connected components.Each stage of the construction adds at most finitely many edges, and thus at each stage of the construction all but finitely many vertices are isolated.
For each e, our plan is the following.We monitor the action of W X e for all oracles X until we see a sufficient measure of X 's produce enough vertices (in a sense to made precise).Then, the idea is to satisfy the requirement by adding edges to these vertices in a way that defeats about half (in the measure-theoretic sense) of the oracles X .This is done by a two-step process.Requirement R e acts by either type I or type II actions, the second type following the first type.In a type I action, R e locks some finite number of vertices, thereby preventing lower priority requirements from adding edges to these locked vertices.In a type II action, R e merges finitely many of G's connected components into one connected component by adding some new edges while maintaining that G is a bipartite graph.This merging is made in a way which ensures that for a sufficient measure of oracles X , W X e is inhomogeneous for the resulting graph.
We now present the construction in full detail.At stage s, we say that − R e requires type I attention if R e has no vertices locked and there are strings of length s witnessing that µ{X : (∃x ∈ W X e,s )(x is not connected to any v locked by R k for any k < e)} > 0.9; − R e requires type II attention if it currently has locked vertices due to a type I action, has never acted according to type II, and there are strings of length s witnessing that µ{X : (∃ y ∈ W X e,s )( y is not connected to any v locked by R k for any k ≤ e)} > 0.9; − R e requires attention if R e requires type I attention or requires type II attention.
At stage 0, E = , and no requirement has locked any vertices.
At stage s + 1, let e < s be least such that R e requires attention (if there is no such e, then go on to the next stage).If R e requires type I attention, let x 0 , x 1 , . . ., x n−1 be vertices that are not connected to any v locked by R k for any k < e and such that the strings of length s witness that µ{X : (∃i < n)(x i ∈ W X e,s )} > 0.9.R e locks the vertices x 0 , x 1 , . . ., x n−1 .All requirements R k for k > e unlock all of their vertices.
If R e requires type II attention, let y 0 , y 1 , . . ., y m−1 be vertices that are not connected to any v locked by R k for any k ≤ e and such that the strings of length s witness that µ{X : (∃ j < m)( y j ∈ W X e,s )} > 0.9.Let x 0 , x 1 , . . ., x n−1 be the vertices that are locked by R e .First we merge the connected components of the x i 's into a single connected component and the connected components of the y j 's into a single connected component.To do this, let a, b, c, d > s be fresh vertices, and add the edges (a, b) and (c, d).The graph is currently bipartite, so for each i < n add either the edge (x i , a) or (x i , b) so as to maintain a bipartite graph.Similarly, merge the connected components of the y j 's by adding either the edge ( y j , c) or ( y j , d) for each j < m.The component of the x i 's is disjoint from the component of the y j 's because the y j 's were chosen not to be connected to the x i 's.Thus both the graph G 1 obtained by adding the edge (a, c) and the graph G 2 obtained by adding the edge (a, d) are bipartite.Each pair {x i , y j } is homogeneous for exactly one of G 1 and G 2 , and the strings of length s witness that µ{X : (∃i < n)(∃ j < m)(x i ∈ W X e,s ∧ y j ∈ W X e,s )} > 0.8 and therefore that µ{X : W X e,s is either G 1 -inhomogeneous or G 2 inhomogeneous} > 0.8.
Thus the strings of length s either witness that µ{X : W X e,s is G 1 -inhomogeneous} > 0.4, in which case we extend to G 1 by adding the edge (a, c), or that µ{X : W X e,s is G 2 -inhomogeneous} > 0.4, in which case we extend to G 2 by adding the edge (a, d).This completes the construction.
To verify the construction, we first notice that G is bipartite because it is bipartite at every stage.
Furthermore, G is recursive because if an edge (u, v) is added at stage s, either u > s or v > s.Thus to check whether an edge (u, v) is in G, it suffices to check whether the edge has been added by stage max(u, v).
We now verify that every requirement is satisfied.Suppose that R e acts according to type II at some stage s + 1.Then R e is satisfied because we have ensured that µ{X : W X e is G-inhomogeneous} > 0.4 and thus that µ{X : W X e is G-homogeneous} ≤ 0.6.
We prove by induction that, for every e ∈ ω, R e is satisfied and there is a stage past which R e never requires attention.Consider R e .If µ{X : W X e is infinite} ≤ 0.9, then R e is satisfied and R e never requires attention.So assume that µ{X : W X e is infinite} > 0.9.By induction, let s 0 be a stage such that no R k for k < e ever requires attention at a stage past s 0 .If R e has locked vertices at stage s 0 , then these vertices remain locked at all later stages because no higher priority R k ever unlocks them.If R e does not have locked vertices at stage s 0 , then let s 1 ≥ s 0 be least such that the strings of length s 1 witness that R e requires type I attention.Such an s 1 exists because µ{X : W X e is infinite} > 0.9 and because the finite set of vertices that are connected to vertices locked by the R k for k < e have stabilized by stage s 0 .R e then requires and receives type I attention at stage s 1 , and the vertices that R e locks at stage s 1 are never later unlocked.So there is a stage s 1 ≥ s 0 by which R e has locked a set of vertices that are never unlocked.If R e has acted according to type II by stage s 1 , then R e is satisfied and never requires attention past stage s 1 .If R e has not acted according to type II by stage s 1 , let s 2 ≥ s 1 be least such that the strings of length s 2 witness that R e requires type II attention.Such an s 2 exists because µ{X : W X e is infinite} > 0.9 and because, past stage s 1 , no requirement except R e can act to connect a vertex to a vertex locked by an R k for a k ≤ e. R e then requires and receives type II attention at stage s 2 .Hence R e is satisfied, and R e never requires attention at a later stage.This completes the proof.Theorem 6.11.RCA 0 + WWKL RCOLOR 2 .

Corollary 3 .
11 (to the proof of Theorem 3.4).The following statements are equivalent over RCA 0 :

11 u 2
30, and the string 10101 in T corresponds in S to u 1 −u 0

Notation 5 . 4 .
Let G = (V, E) be a graph, let a, b ∈ V , and let ν : V → k be a k-coloring of G.We write a = ν b if ν(a) = ν(b).

Widget 5 . 5 .
Rx → y y →z (a, u) is the following widget.Let ν be a 3-coloring of Rx → y y →z (a, u).If a = ν x then u = ν y, and if a = ν y then u = ν z. (ii) Every 3-coloring of the subgraph of Rx → y y →z (a, u) induced by {x, y, z, a} can be extended to a 3coloring of Rx → y y →z (a, u).(iii)In every 3-coloring of Rx → y y →z (a, u), the color of each vertex in {u, v} determines the color of a.
In the diagram above, the box labeled 'Rx → y y →z (ℓ, r)' represents an Rx → y y →z (ℓ, r) sub-widget.The vertices ℓ and r are the same as those appearing inside Rx → y y →z (ℓ, r).They have been displayed to show how they connect to the rest of the U x, y,z (ℓ, b, u) widget.The vertices x, y, and z are also the same as the corresponding vertices appearing inside Rx → y y →z

Lemma 5. 8 .
(i) Every 3-coloring ν of the subgraph of U x, y,z (ℓ, b, u) induced by {x, y, z, ℓ, b} can be extended to a 3-coloring of U x, y,z (ℓ, b, u).(ii) If ν is a 3-coloring of U x, y,z (ℓ, b, u) in which ℓ = ν x and b = ν y, then u = ν x. (iii) Every 3-coloring ν of the subgraph of U x, y,z (ℓ, b, u) induced by {x, y, z, ℓ, b} in which ℓ = ν x and b = ν y can be extended to a 3-coloring of U x, y,z (ℓ, b, u) in which u = ν z. (iv) Every 3-coloring ν of the subgraph of U x, y,z (ℓ, b, u) induced by {x, y, z, ℓ, b} in which ℓ = ν y can be extended to a 3-coloring of U x, y,z (ℓ, b, u) in which u = ν y.Proof.For (i), let ν be a 3-coloring of the subgraph of U x, y,z (ℓ, b, u) induced by {x, y, z, ℓ, b}.− If ℓ = ν x and b = ν x, then color the widget so that l = ν y, r = ν y, d = ν y, and u = ν z. − If ℓ = ν x and b = ν y, then color the widget so that l = ν y, r = ν y, d = ν z, and u = ν x. − If ℓ = ν x and b = ν z, then color the widget so that l = ν y, r = ν y, d = ν y, and u = ν z. − If ℓ = ν y and b = ν x, then color the widget so that l = ν x, r = ν z, d = ν z, and u = ν y. − If ℓ = ν y and b = ν y, then color the widget so that l = ν x, r = ν z, d = ν x, and u = ν y. − If ℓ = ν y and b = ν z, then color the widget so that l = ν x, r = ν z, d = ν x, and u = ν y.

1
Now build G in stages by considering the clauses in C one-at-a-time.For clauseℓ 0 ∨ℓ 1 ∨• • •∨ℓ n−1 , find the previously appearing clause s 0 ∨ s 1 ∨ • • • ∨ s m−1 having the longest common prefix with

Corollary 5 . 14 .Question 5 . 16 .Theorem 5 . 17 .
the truth value coded by c for every a ∈ H ′′ , so H ′′ is indeed an infinite homogeneous set for C.It follows that RWKL, RCOLOR k , and LRCOLOR k are equivalent for every fixed k ≥ 3.For every k ∈ ω with k ≥ 3, RCA 0 ⊢ RWKL ↔ RCOLOR k ↔ LRCOLOR k .Proof.Fix k ∈ ω with k ≥ 3. RCA 0 ⊢ RWKL → LRCOLOR k by Lemma 5.3, and clearly RCA 0 ⊢ LRCOLOR k → RCOLOR k .It is easy to see that RCA 0 ⊢ RCOLOR k → RCOLOR 3 .Given a locally 3-colorable graph G, augment G by a clique C containing k − 3 fresh vertices, and put and edge between every vertex in C and every vertex in G.The resulting graph G ′ is locally kcolorable, and every infinite set that is k-homogeneous for G ′ is also 3-homogeneous for G. Finally, RCA 0 ⊢ RCOLOR 3 → RWKL by Theorem 5.13.The question of the exact strength of RCOLOR 2 remains open.We were unable to determine if RCOLOR 2 implies RWKL or even if RCOLOR 2 implies DNR.Question 5.15.Does RCA 0 ⊢ RCOLOR 2 → RWKL?Does RCA 0 ⊢ RCOLOR 2 → DNR?However, we are able to show that RCOLOR 2 and LRCOLOR 2 are equivalent.RCA 0 ⊢ RCOLOR 2 ↔ LRCOLOR 2 .