Convex geometries are extremal for generalized Sauer-Shelah bound

. Sauer-Shelah lemma provides an exact upper bound on the size of set families with bounded Vapnik-Chervonekis dimension. When applied to lattices represented as closure families, this lemma can be used to describe a class of extremal lattices as those obtaining this bound. Here we give an alternative deﬁnition of convex geometries as extremal objects of generalized Sauer-Shelah bound.


Introduction
Sauer-Shelah lemma is a result from extremal set theory, establishing an exact bound on the number of sets in a family, not shattering any k-set.
Lemma 1 (Sauer-Shelah). If F is a family of subsets of U , |U | = n, and |F| > H(n, k) , where H(n, k) is a sum of first k binomial coefficients of n, that is, then F shatters some k-set.
The above bound is exact, as it is trivially reached by the family of all subsets of size at most k − 1. In [1] A.Albano and the author characterized, although in lattice-theoretic terms, the closure families obtaining Sauer-Shelah bound.
The natural extension of this bound is to consider families which do not shatter any set from some fixed antichain A, the original bound corresponding to the antichain A k of all k-sets. Indeed, Sauer-Shelah lemma can be easily extended to this setting.
In this paper we consider the problem of describing closure families which are extremal with regard to this extended bound. As it turns out, what we get is exactly convex geometries.
In Section 2 we introduce basic terminology. In Section 3 we give a generalized version of Sauer-Shelah lemma and give a definition of extremal closure family. In Section 4 we give intermediary results used later for characterization of extremal families, and define extremal lattices. In Section 5 we characterize extremal closure families as convex geometries and extremal lattices as meetdistributive lattices. In Section 6 we reformulate circuit characterization of antimatroids in our terms in order to obtain characterization of extremal closure families in terms of implications. Finally, in Section 7 we consider a those problem of characterization of those antichains, which we call extremal, for which generalized Sauer-Shelah bound is exact on closure families. As we show, this problem turns out to be NP-complete.

Preliminary definitions
If not stated otherwise, all objects wee are dealing with in our paper are finite. Typically, we are dealing with subsets of a base set U , |U | = n. A k-set is a set with k elements. For a set X ⊆ U and a set family F, we define a trace of F on X, denoted T F (X), as We denote the size of T F (X) by t F (X), also we omit the subscript if the set family is clear from the context. The family of all subsets of X is denoted by P(X). We say that F shatters X if T F (X) = P(X). The Vapnik-Chervonekis (VC) dimension of F is the maximal size of a set, shattered by F.
A family I is hereditary if it contains every subset of A, for any A in I. Family C is a closure system if it contains U and is closed under set intersection. When dealing with closure systems we will denote by X the least closed subset containing X. Every closure system C is a lattice by set inclusion which we denote by L(C), or simply L C .
For a lattice L and an element x ∈ L we denote by J(L) the set of join irreducible elements of L, and define J(x) = (x] ∩ J(L). It is common to identify L with closure system F L on J(L) defined by We denote boolean lattice with k atoms by B(k). We say that a lattice L is B(k)-free if it does not have an order-embedding of B(k). This is equivalent to the family F L not to shatter a k-set, or, otherwise stated, to be of VC-dimension at most k − 1.
For a lattice L, a set S ⊆ J(L) is called a minimal generator if for all x in J.
In [1] it was shown that it is possible to construct a closure system C of size H(n, k) not shattering any k-set, which is equivalent to the lattice L C being B(k)-free. These lattices were called (n, k)-extremal and were completely characterized in [1] as lattices obtained by recursive application of a specific doubling construction. Thus, (n, k)-extremal lattices can be characterized as lattices with VC-dimension at most k − 1, reaching Sauer-Shelah bound.
As we show below, taking the latter as a definition but for a generalization of Sauer-Shelah bound, we obtain exactly convex geometries, or, in latticetheoretical terms, meet-distributive lattices.
Let A be an antichain of sets on U . We say that F does not shatter A if it does not shatter any set A from A. By I(A) we denote the family of all subsets of U not containing any set of A as a subset, that is and by s(A) we denote the size of I(A).
For two antichains A and B we say that A refines B, denoted A B, if for each b ∈ B there is a ∈ A such that a ≤ b. In particular, A B whenever B ⊆ A. The refinement relation is a partial order on antichains, see for example Lemma 1.15 in [5]. The following proposition is straightforward We need the following lemma: Lemma 2 (Frankl). For any family A there is a hereditary family I of equal size such that t S (I) ≤ t S (A) for any S ⊆ U .
Proof. This statement used implicitly in [3]. In its present state it is formulated as Theorem 10.2 in [7]. We say that set family F is extremal for an antichain A if |F| = s(A) and if F does not shatter A. We say that F is extremal if it is extremal for some antichain A. For any family F in U , not necessarily extremal, we define a blocking antichain A F of F as a family of all subsets of U , minimal with respect to being not shattered by F. Trivially, A F is an antichain. Note also, that I(A F ) is the family of all subsets, shattered by F. Proof. Let J be a hereditary family. We claim that in this case J is extremal for A J . Indeed, J does not shatter any set from A J , and suppose that |J | < s(A J ) = |I = I(A J )|. As I is a maximal of all hereditary sets not shattering A J , we have J I, and we can take a minimal set B in I − J . But, from minimality of B it follows that B − x ∈ J , for all x ∈ J . Thus, J does not shatter B, but shatters B − x, for any x ∈ B, which means that B ∈ A. But, as B ∈ I, it follows that I shatters A, a contradiction. Proof. From the definition of A F it follows that for any B ∈ B there is A ∈ A F with A ⊆ B, that is, A F refines B. From Proposition 1 it follows that |F| ≤ s(A F ) < s(B) whenever B = A F , which contradicts extremality.
Lemma 5. If F is extremal then F almost shatters every set in A F .
Proof. By Lemma 2 there is a hereditary family I of size |F| such that t I (A) ≤ t F (A) for all A ∈ A F , in particular I does not shatter A F . Due to extremality of F, there is exactly one such hereditary family, namely It is not hard to construct a counterexample of Lemma 3 for arbitrary family F. For example, let F be a family of sets in U = {1, 2, 3, 4} not covering 123, and intersecting 234, that is Let us consider some trivial cases of extremal families. For any set U , there is a smallest with regards to refinement antichain A 0 = {∅}, and the empty family of subsets F 0 is the only family not shattering A 0 , also, F 0 is extremal for A 0 . Notice, that A 0 is the only antichain containing the empty set. Apart from it A 0 is the only antichain in a set ∅, and F 0 is the only set family in ∅, which is both hereditary, closure system, and extremal for A 0 .

Reduction of antichains and closure systems
We call an antichain A redundant if in A there is a one-point set, and irredundant otherwise. For an antichain A we define reduction of A, denoted R(A), as an antichain We say that a set family F on X is redundant if there is x ∈ X such that t F (x) = 1. We define reduction of F, denoted R(F) as a family For closure systems we introduce additional notation. We say that a closure system C on U is ambiguous if x = X for some x ∈ U R and X ⊆ U R − x, otherwise we call C unambiguous. In terms of lattices, for L = L C , irredundant elements of C are those mapped to 0 L by closure operator, and C is ambiguous if either two distinct irredundant elements in U are mapped to the same element in L, or some irredundant element in U is mapped to a join reducible element.
The following easy proposition establishes a correspondence between reduction of antichains and reduction of set families.
Proposition 2. For a set family F extremal for an antichain A, holds Moreover, F can be chosen hereditary (closure system) whenever F is hereditary (closure system).

Proposition 3.
An element x is redundant for a closure system C iff x ∈ ∅.
Proposition 2, in particular, implies that an extremal closure family can be redundant. On the other hand, it cannot be ambiguous. Lemma 6. If closure family C is extremal then C is unambiguous.
Proof. Suppose C is ambiguous and let a = A for some a ∈ U R and A ⊆ U R − a. Let us define an extended closure system C A as Trivially, C A is a closure system and it is a proper extension of C, as A ∈ C A − C.
We claim that C and C A shatter same sets. Indeed, if X is shattered by C then it is shattered by C A as C ⊂ C A . On the other hand, let C A shatter X, and let us take  Proof. (⇒) : Let L be a lattice of closed sets of C on U and let us denote by ϕ : C → L the bijection between L and C. As C is irredundant, t C (x) = 2, for every x ∈ U . Thus, there is X x ∈ C such that x ∈ X x and consequently On the other hand, as C is unambiguous, is a bijection between U and J(L) and C ∼ = J(L).
x either belongs to J(y) for every y ∈ L, which is impossible because x ∈ J(0) or x does not belong to any J(y), which is impossible because Similarly, if C is ambiguous then there is x ∈ J(L) such that x = J(x) = X, for X ⊆ J(L) − x. But then X = J ( (X)), which is only possible if X = x, a contradiction.
We say that a lattice L shatters A ⊆ J(L) if F L shatters A. We say that L is extremal if F L is extremal, in which case we denote A(F L ) by A L . Example 1. A lattice with a single element is extremal with F L being an empty family on an empty set of join irreducible elements. As mentioned earlier, A L = A 0 .
Example 2. Let L be a distributive lattice represented as a family of order-ideals over a poset P . Then L is extremal and A L is an antichain over P given by

Characterization of extremal lattices
In our definition of meet-distributive lattices we follow [4], namely, a lattice L is meet-distributive if an interval [x, y] is a boolean lattice whenever x is a meet of elements covered by y, for any y ∈ L.
A set X ∈ J(L) is called an irredundant join representation of a = X if (X − x) < a, for every x ∈ X. We denote by JIR the family in J(L) of all irredundant join representations in L.
In a finite lattice every element has at least one (possibly more) irredundant join representations. On the other hand, no two distinct elements share an irredundant join representation. Thus, |L| ≤ |JIR(L)|.
We refer to Theorem 44 of [6] for the following fact: Lemma 8. A finite lattice L is meet-distributive if and only if every element of L has a unique irredundant join representation.
The following two lemmas are straightforward and relate irredundant join representations with extremality.
Lemma 9. For every lattice L, the family JIR(L) is hereditary.
Proof. Let X be an irredundant join representation and suppose X−x ∈ JIR(L), for some x ∈ X. This means that (X − x) = (X − {x, y}), for some y ∈ X, y = x. But then For any X ⊆ A let us take X ∈ F L . If X = X ∩ A X then X = X and thus X ∈ JIR(L) which is impossible as X ⊂ A ∈ JIR(L)) and JIR(L) is hereditary. Thus, for every X ⊆ A holds X ∩ A = X and F L shatters A.
A convex geometry is a base set U endowed with a closure operator ϕ : X → X satisfying the anti-exchange property: x ∈ A ∪ y implies y ∈ A ∪ x, for all closed A, x, y ∈ A, x = y. Trivially, every convex geometry is a closure system. As it was shown by Edelman, convex geometries are closely related to meet-distributive lattices, see Theorem 3.3 in [4].
Lemma 11. A lattice L is a lattice of closed sets of convex geometry if and only if L is meet-distributive.
Notice that Lemma 11 works in one direction only, that is, if a lattice of a closure system C is meet-distributive, it does not imply that C is a convex geometry, for example let C in U = {1, 2} be Then L C ∼ = B(1), which is meet-distributive. On the other hand 1, 2 ∈ ∅, but 2 ∈ 1 and 1 ∈ 2.
We use earlier developed notions of reduction and unambiguity to draw a more detailed connection between convex geometries and meet-distributive lattices.
Lemma 12. Closure system C is a convex geometry iff R(C) is.
Proof. By Proposition 3, an element x is redundant if and only if x ∈ ∅. Thus, in anti-exchange property, we can consider only irredundant elements x and y, and the statement of the lemma follows.
Lemma 13. If C is a convex geometry then it is unambiguous.
Proof. Suppose otherwise and take x ∈ U R and X ⊆ (U R − x) such that x = X. Let Y = X − x, then Y = X, in particular, Y is not closed. Let A be any maximal closed set in Y . As Y is not closed it follows that A Y and we can chose y ∈ Y − A. As On the other hand, y ∈ X = x, hence y ∈ A ∪ x, a contradiction.
As a consequence we might formulate the following characterization of extremal lattices. This means that L is extremal iff |L| = |I(A)| = |JIR(L)|, however this holds iff every element of L has a unique irredundant join representation. By Lemma 8 this happens iff L is meet-distributive.
(2) : By Lemma 12 and Proposition 2 it is enough to establish a correspondence for irredundant C, and by Lemma 13 and Lemma 6 we can also consider only unambiguous C. By Lemma 7, irredundant and unambiguous C corresponds to F L , for some lattice L. Lemma 11 and the definition of extremality of lattices show that convex geometries and extremal closure systems correspond to meetdistributivity and extremality of corresponding lattices, and their equivalence was proven in part (1).

Convex geometries and implications
We now aim at a characterization of extremal closure systems in terms of implications. The following notation is from [6].
T . An implication holds in a family F if every set of F respects it. Given a family of implications I, we denote by C(I) a family of sets respecting all implications in I. It is evident that in this case C(I) is a closure system.
Lemma 14. For an extremal closure system C and any A ∈ A C there is unique element a ∈ A such that an implication (A − a) → a holds in F.
Proof. By Lemma 5 there is exactly one set X in P(A) − T C (A). Notice also that T C (A) is a closure system, and thus |X| = |A − 1|, otherwise X can be represented as an intersection of two subsets of size |X| + 1. Thus, X = A − a for some a ∈ A. Now, from the definition of trace it follows that for any C ∈ C, a ∈ C whenever (A − a) ⊆ C, that is, C respects implication (A − a) → a.
Following [8], we define a rooted set as a pair (X, x), x ∈ X, where x is called root. We also define rooted antichain as a family of rooted sets A R = {(A i , a i )} such that family A = {A i } is an antichain. In this case we call A corresponding antichain of A R and A R corresponding rooted antichain of A. For simplicity, we distinguish corresponding antichain and rooted antichain by upper index R. In what follows we identify a rooted set (A, a) with an implication (A − a) → a. We say that a rooted antichain A R is extremal if C(A R ) is.
Using Lemma 14, for a given extremal closure system C we may uniquely construct a rooted antichain A R C such that A C is the extremal set of C and such that C respects A R C . We call A R C rooted blocking antichain of C.
Lemma 15. For an extremal closure family C, it holds C = C(A R C ).
Proof. Obviously, C respects A R C , thus C ⊆ C(A R C ). On the other hand, C(A R C ) does not shatter any A ∈ A, thus |C| ≥ |C(A R C )|.
It turns out that rooted antichains which give rise to extremal closure families may be easily described using characterization of circuits of antimatroids from [2]. Proof. Theorem 2 reveals extremal closure systems as convex geometries, which are turned into antimatroids by complementation. In the same time, rooted sets from the rooted blocking antichain of an extremal closure family C correspond to rooted circuits of complemented antimatroid C * , see Lemma 4.4 in [8]. Now, statement of the theorem follows directly from the characterization of rooted families yielding antimatroids given in Theorem 7 in [2].

Identification of extremal antichain is NP-complete
We say that an antichain A is extremal, if it is extremal for some closure family C, or, equivalently, if it is a corresponding antichain for some extremal rooted antichain. Notice that the family I(A) is extremal for any A, thus the condition for C to be a closure family is essential.
As it was mentioned before, in [1] it was shown that it is possible to construct extremal closure system for the antichain A k of all k-sets of U , for any k, thus all antichains A k are extremal. In general, however, it is not easy to say whether a given antichain is extremal, in fact, we show below that this problem is NPcomplete.
Lemma 16. For a given antichain A there is an NP algorithm deciding whether A is extremal.
Proof. Given an antichain A, the described algorithm first nondeterministically guesses a rooted antichain A R , corresponding to A, and then checks whether A R is extremal using characterization from Theorem 3, which demands time cubic with respect to the number of sets in A.
We call a (rooted) antichain A intransitive if for any distinct A and B and for any point We say that a rooted antichain A R is accordant if every point from its base set is either the root for all sets from A R containing it, or for none of them.

Lemma 17. An intransitive rooted antichain is extremal if and only if it is accordant.
Proof. If a rooted antichain is accordant, then a ∈ B − b, for any rooted sets (A, a) and (B, b) and by Theorem 3 it is rooted, proving one side of the statement. Now, let A R be extremal intransitive rooted antichain. Suppose A R is not accordant, that is, there is a point a and two rooted sets (A, a) and (B, b) in A R , a ∈ B − b. Then by Theorem 3 there is (C, b) ∈ A R such that C ∈ A ∪ B − a, but this contradicts the intransitivity of A R .
We denote by R(A R ) the set of roots of A R , that is, R(A R ) = {a | (a, A) ∈ A R }. For an antichain A we define a root set S as a set such that |S ∩ A| = 1, for every A ∈ A, and such that every x ∈ S lies in some set of A. Then for such A and S we can define accordant corresponding rooted antichain A R (S) by setting root of A to a = A ∩ R. Then S = R(A R (S)). On the other hand, R(A R ) is a root set for every accordant rooted antichain A R .
Lemma 18. An intransitive antichain is extremal if and only it has root set.
Proof. By Lemma 17, an intransitive antichain is extremal if and only if it has corresponding accordant rooted antichain. The statement of the lemma now follows by observing that accordant rooted antichains trivially correspond to root sets.
We will prove NP-hardness of antichain extremality by polynomial reduction of 3-SAT to this problem. Let us recall that 3-SAT is a problem of determining whether a given boolean formula F in conjunctive normal form (CNF) with each clause containing exactly three literals is satisfiable. Let us recall that a boolean formula in CNF is a conjunction of clauses, each clause is a disjunction of literals and each literal is either a variable or its negation.
We utilize simplified notation for partial valuations on the set of variables, for example we write abc for a partial valuation that assigns True to a and c and False to b. For a clause cl (partial valuation α) we denote by V (cl) (by V (α)) the set of variables of cl (of α). Thus, for a 3-SAT formula F , |V (cl)| = 3 for each clause cl of F .
For a partial valuation α and a variable x ∈ V (α) we denote by α(x) the value of α on x, and we denote by x : α(x) the partial valuation that assigns value α(x) to variable x. Thus, abc(b) = F alse and b : abc(b) = b. We say that a partial valuation α is compliant with a complete valuation ϕ if α(x) = ϕ(x) for every x ∈ V (α).
Theorem 4. Problem of determining whether given antichain is extremal is NP-hard.
Proof. Let F be considered 3-CNF formula, V the set of its variables and C the set of its clauses, each consisting of three literals. We are going to construct an antichain which is extremal if and only if F is satisfiable.
Let us construct antichain A F . We use partial valuation, sometimes with indexes, as elements of base set U of Intuitively, points in U var describe valuation of variables of F and points in U cl describe values of clauses under given valuation.
The antichain A F consists of four parts, For example, for formula F * = x ∨ y with a single clause, U * and A * F look as follows:   Thus constructed, antichain A F is polynomial with respect to the size of F , also, A F is intransitive. We claim that A F is extremal if and only if F is satisfiable. By Lemma 18, the latter is true if and only if there is a root set for A.
We claim that S is a root set of A F . Obviously, for any {x, x} ∈ A var , ϕ is compliant with exactly one of x and x, thus S ∩ {x, x} = 1, for any variable x. For any clause c of F , ϕ satisfies c, thus there is exactly one α ∈ T (c), compliant with ϕ and a unique intersection of S with A = T (c), for any A ∈ A cl .
Finally, for any A = {α v,2 , v : α(v)} ∈ A 2 link , either ϕ(v) = α(v) and v : α(v) ∈ S, or ϕ(v) = α(v), in which case α v,2 ∈ S. Figure 2 below shows rooted extremal antichain corresponding to our exemplary formula F * = x ∨ ¬y with root set S * constructed for valuation ϕ * = xy. Conversely, let F be unsatisfiable, and let R be a root set for A F . Then for every x ∈ V exactly one of x and x lies in R, and let us define complete valuation ϕ by ϕ(x) = T rue if x ∈ R and ϕ(x) = F alse otherwise. As F is unsatisfiable, there is a clause c ∈ C such that ϕ(c) = F alse.
Let α be a complete valuation of V (c) which is the root of T (c) ∈ A 1 link . Then α satisfies c and consequently, α is not compatible with ϕ.
Then there is a variable x ∈ V (c) such that x : α(x) = ϕ(x), by the definition of ϕ this means that x : α(x) ∈ S. But as S has a common point with {α x,2 , x : α(x)} ∈ A 2 link , α x,2 ∈ S. But the both points α x,2 and α lie in S ∩ A for A = {α, α x,2 , α x,2 } ∈ A 1 link , a contradiction. From Figure 3 below one can see that a root set cannot be chosen for a valuation that falsifies the only clause of F * . Theorem 4 together with Lemma 16 prove NP-completeness of the problem of determining whether given antichain is extremal.