Hypergraph Tur´an densities can have arbitrarily large algebraic degree

Grosu [Journal of Combinatorial Theory (B), 118 (2016) 137–185] asked if there exist an integer r ≥ 3 and a ﬁnite family of r -graphs whose Tur´an density, as a real number, has (algebraic) degree greater than r − 1. In this note we show that, for all integers r ≥ 3 and d , there exists a ﬁnite family of r -graphs whose Tur´an density has degree at least d , thus answering Grosu’s question in a strong form.


Introduction
For an integer r ≥ 2, an r-uniform hypergraph (henceforth, an r-graph) H is a collection of r-subsets of some finite set V .Given a family F of r-graphs, we say H is F-free if it does not contain any member of F as a subgraph.The Turán number ex(n, F) of F is the maximum number of edges in an F-free r-graph on n vertices.The Turán density π(F) of F is defined as π(F) := lim n→∞ ex(n, F)/ n r ; the existence of the limit was established in [12].The study of ex(n, F) is one of the central topic in extremal graph and hypergraph theory.For the hypergraph Turán problem (i.e. the case r ≥ 3), we refer the reader to the surveys by Keevash [13] and Sidorenko [18].
For r ≥ 3, determining the value of π(F) for a given r-graph family F is very difficult in general, and there are only a few known results.For example, the problem of determining π(K r ℓ ) raised by Turán [19] in 1941, where K r ℓ is the complete r-graph on ℓ vertices, is wide open and the $500 prize of Erdős for solving it for at least one pair ℓ > r ≥ 3 is still unclaimed.
For r = 2 the celebrated Erdős-Stone-Simonovits theorem [6,7] determines the Turán density for every family F of graphs; in particular, it holds that Π (2) The problem of understanding the sets Π (r) fin and Π (r) ∞ of possible r-graph Turán densities for r ≥ 3 has attracted a lot of attention.One of the earliest results here is the theorem of Erdős [5] from the 1960s that Π (r) ∞ ∩ (0, r!/r r ) = ∅ for every integer r ≥ 3.However, our understanding of the locations and the lengths of other maximal intervals avoiding r-graph Turán densities and the right accumulation points of Π (r) ∞ (the so-called jump problem) is very limited; for some results in this direction see e.g.[1,8,9,17,21].
fin (and thus the former set is easier to understand) and that Π (r) ∞ has cardinality of continuum (and thus is strictly larger than the countable set Π (r) fin ), see respectively Proposition 1 and Theorem 2 in [16].
For a while it was open whether Π (r) fin can contain an irrational number (see the conjecture of Chung and Graham in [3, Page 95]), until such examples were independently found by Baber and Talbot [2] and by the second author [16].However, the question of Jacob Fox ([16, Question 27]) whether Π Grosu [11] initiated a systematic study of algebraic properties of the sets Π Recall that the (algebraic) degree of a real number α is the minimum degree of a non-zero polynomial p with integer coefficients that vanishes on α; it is defined to be ∞ if no such p exists (that is, if the real α is transcendental).In the same paper, Grosu [11,Problem 3] posed the following question.
Problem 1.1 (Grosu).Does there exist an integer r ≥ 3 such that Π (r) fin contains an algebraic number α of degree strictly larger than r − 1?
Apparently, all r-graph Turán densities that Grosu knew or could produce with his machinery had degree at most r−1, explaining this expression in his question.His motivation for asking this question was that if, on input F, we can compute an upper bound on the degree of π(F) as well as on the absolute values of the coefficients of its minimal polynomial, then we can compute π(F) exactly, see the discussion in [11,Page 140].
In this short note we answer Grosu's question in the following stronger form.Theorem 1.2.For every integer r ≥ 3 and for every integer d there exists an algebraic number in Π (r) fin whose minimal polynomial has degree at least d.
Our proof for Theorem 1.2 is constructive; in particular, for r = 3 we will show that the following infinite sequence is contained in Π

Preliminaries
In this section, we introduce some preliminary definitions and results that will be used later.
For an integer r ≥ 2, an (r-uniform) pattern is a pair P = (m, E), where m is a positive integer, E is a collection of r-multisets on [m] := {1, . . ., m}, where by an r-multiset we mean an unordered collection of r elements with repetitions allowed.Let V 1 , . . ., V m be disjoint sets and let ) consist of all r-subsets of V whose profile is S. We call this r-graph the blowup of S and the r-graph is called the blowup of E (with respect to V 1 , . . ., V m ).We say that an r-graph H is a P -construction if it is a blowup of E. Note that these are special cases of the more general definitions from [16].
It is easy to see that the notion of a pattern is a generalization of a hypergraph, since every r-graph is a pattern in which E is a collection of (ordinary) r-sets.For most families F whose Turán problem was resolved, the extremal F-free constructions are blowups of some simple pattern.For example, let P B := (2, {{{1, 2, 2}}, {{ 1, 1, 2 }}}), where we use {{ }} to distinguish multisets from ordinary sets.Then a P B -construction is a 3-graph H whose vertex set can be partitioned into two parts V 1 and V 2 such that H consists of all triples that have nonempty intersections with both V 1 and V 2 .A famous result in the hypergraph Turán theory is that the pattern P B characterizes the structure of all maximum 3-graphs of sufficiently large order that do not contain a Fano plane (see [4,10,14]).
For a pattern P = (m, E), let the Lagrange polynomial of E be , where E(i) is the multiplicity of i in the r-multiset E. In other words, λ E gives the asymptotic edge density of a large blowup of E, given its relative part sizes x i .
The Lagrangian of P is defined as follows: Since we maximise a polynomial (a continuous function) on a compact space, the supremum is in fact the maximum and we call the vectors in ∆ m−1 attaining it P -optimal.Note that the Lagrangian of a pattern is a generalization of the well-known hypergraph Lagrangian that has been successfully applied to Turán-type problems (see e.g.[1,9,20]), with the basic idea going back to Motzkin and Straus [15].
For i ∈ [m] let P − i be the pattern obtained from P by removing index i, that is, we remove i from [m] and delete all multisets containing i from E (and relabel the remaining indices to form the set [m − 1]).We call P minimal if λ(P − i) is strictly smaller than λ(P ) for every i ∈ [m], or equivalently if no P -optimal vector has a zero entry.For example, the 2-graph pattern P := (3, { {{1, 2}}, {{1, 3 }} }) is not minimal as λ(P ) = λ(P − 3) = 1/2.
In [16], the second author studied the relations between possible hyrergraph Turán densities and patterns.One of the main results from [16] is as follows.
Theorem 2.1 ( [16]).For every minimal pattern P there exists a finite family F of rgraphs such that π(F) = λ(P ), and moreover, every maximum F-free r-graph is a Pconstruction.
The following observation follows easily from the definitions.
Observation 2.2.If P is a minimal pattern, then P + s is a minimal pattern for every integer s ≥ 1.
For the Lagrangian of P + s we have the following result.
Proposition 2.3.Suppose that r ≥ 2 is an integer and P is an r-uniform pattern.Then for every integer s ≥ 1 we have In particular, the real numbers λ(P + s) and λ(P ) have the same degree.
Let x := 1 s m+s i=m+1 x i and note that m i=1 x i = 1 − sx.Since λ E is a homogenous polynomial of degree r, we have This and the AM-GM inequality give that For x ∈ [0, 1/s], the function (1 − sx) r (rx) s , as the product of s + r non-negative terms summing to r, is maximized when all terms are equal, that is, at x = 1 r+s .So To prove the other direction of this inequality, observe that if we take (x 1 , . . ., x m ) = r r+s (y 1 , . . ., y m ), where (y 1 , . . ., y m ) ∈ ∆ m−1 is P -optimal, and take r+s , then all inequalities above hold with equalities.
3 Proof of Theorem 1.2 In this section we prove Theorem 1.2.By Theorem 2.1, it suffices to find a sequence of r-uniform minimal patterns (P k ) ∞ k=1 such that the degree of the real number λ(P k ) goes to infinity as k goes to infinity.Furthermore, by Observation 2.2 and Proposition 2.3, it suffices to find such a sequence for r = 3.So we will assume that r = 3 in the rest of this note.
Figure 1: Constructions with one level and two levels.
To start with, we let P 1 := (3, {{{1, 2, 3}}, {{1, 3, 3}}, {{2, 3, 3}}}).Recall that a 3-graph H is a P 1 -construction (see Figure 1) if there exists a partition that the edge set of H consists of (a) all triples that have one vertex in each V i , (b) all triples that have one vertex in V 1 and two vertices in V 3 , and (c) all triples that have one vertex in V 2 and two vertices in V 3 .
The pattern P 1 was studied by Yan and Peng in [20], where they proved that there exists a single 3-graph whose Turán density is given by P 1 -constructions which, by λ(P 1 ) = 1/ √ 3, is an irrational number.It seems that some other patterns could be used to prove Theorem 1.2; however, the obtained sequence of Turán densities (i.e. the sequence in (1)) produced by using P 1 is nicer than those produced by the other patterns that we tried.
Proof.We use induction on k where the base k = 0 is easy to check directly (or can be derived by adapting the forthcoming induction step to work for k = 0).Let k ≥ 1.
To prove the other direction of this inequality, one just need to observe that when we choose where (y 1 , . . ., y 2k+1 ) ∈ ∆ 2k is a P k -optimal vector, then all inequalities above hold with equality.Therefore, λ(P k+1 ) = 1/ 3 − 2λ(P k ).
In order to finish the proof of Theorem 1.2 it suffices to prove that the degree of µ k := λ(P k ) goes to infinity as k → ∞.This is achieved by the last claim of the following lemma.Lemma 3.2.Let p 1 (x) := 3x 2 − 1 and inductively for k = 1, 2, . . .define Then the following claims hold for each k ∈ N : Let us turn to Part (c).The relation in (4) when taken modulo 3 reads that Thus, c k+1,j ≡ c k,2 k −j/2 (mod 3) for all even j between 0 and 2 k+1 , while c k+1,j ≡ 0 (mod 3) for odd j (in fact, c k+1,j = 0 for all odd j since p k+1 is an even function).In terms of the sequences (b ℓ,j ) 2 ℓ j=0 , this relation states that b k+1,j ≡ b k,j/2 (mod 3) for all even j with 0 ≤ j ≤ 2 k , while b k+1,j ≡ 0 (mod 3) for all odd j.This implies Part (c.i).For Part (c.ii), the relation in (4) when taken modulo 9 gives that c k+1,0 ≡ c k,2 k and c k+1, Since c k,1 is divisible by 3, we have in fact that c k+1,2 k+1 ≡ c k,0 • 2 2 k ≡ c k,0 (mod 9).By the induction hypothesis, this implies that 9 does not divide b k+1,0 .
By the argument above, c k+1,2 k+1 is non-zero module 3 for odd k and non-zero module 9 for even k.Thus, regardless of the parity of k, the degree of the polynomial p k+1 is exactly 2 k+1 .Moreover, p k+1 satisfies Eisenstein's criterion for prime q = 3 (namely, that q divides all coefficients, except exactly one at the highest power of x or at the constant term while the other of the two is not divisible by q 2 ).By the criterion (whose proof can be found in e.g.[16,Section 4]), the polynomial p k+1 is irreducible, proving Part (d).
By putting the above claims together, we see that µ k+1 is a root of an irreducible polynomial of degree 2 k+1 , establishing Part (e).This completes the proof the lemma (and thus of Theorem 1.2)

Concluding remarks
Our proof of Theorem 1.2 shows that for every integer d which is a power of 2 there exists a finite family F of r-graphs such that π(F) has algebraic degree d.It seems interesting to know whether this is true for all positive integers.fin whose algebraic degrees are not powers of 2. For example, the pattern ([3], {{1, 2, 3}}, {1, 2}) with recursive parts 1 and 2 (where we can take blowups of the single edge {{1, 2, 3}} and recursively repeat this step inside the first and the second parts of each added blowup) gives a Turán density in Π (3) fin (by [16,Theorem 3], a generalisation of Theorem 2.1) whose degree can be computed to be 3.However, we did not see any promising way of how to produce a pattern whose Lagrangian has any given degree d.
can contain a transcendental number remains open.
proved a number of general results that, in particular, directly give further examples of irrational Turán densities.

3 . 3 −
(a) p k (µ k ) = 0; (b) p k is a polynomial of degree at most 2 k with integer coefficients: p k (x) = 2 k i=0 c k,i x i for some c k,i ∈ Z; (c) the integers b k,i := c k,i for even k and b k,i := c k,2 k −i for odd k satisfy the following: (c.i) for each integer i with 0 ≤ i ≤ 2 k , 3 divides b k,i if and only if i = 2 k ; (c.ii) 9 does not divide b k,0 ; (d) the polynomial p k is irreducible of degree exactly 2 k ; (e) the degree of µ k is 2 k .Proof.Let us use induction on k.All stated claims are clearly satisfied for k = 1, when p 1 (x) = 3x 2 − 1 and µ 1 = 1/ √ Let us prove them for k + 1 assuming that they hold for some k ≥ 1.For Part (a), we have by Proposition 32µ k ) − 1 2/(3 − 2µ k ) = µ k and thus p k+1 (µ k+1 ) = (2µ 2 k+1 ) 2 k p k (µ k ), which is 0 by induction.Part (b) also follows easily from the induction assumption:

Problem 4 . 1 .
Let r ≥ 3 be an integer.Is it true that for every positive integer d there exists a finite family F of r-graphs such that π(F) has algebraic degree exactly d?By considering other patterns, one can get Turán densities in Π (r)