Eckhoff’s problem on convex sets in the plane

Eckhoff proposed a combinatorial version of the classical Hadwiger–Debrunner (p, q)-problems as follows. Let F be a finite family of convex sets in the plane and let m > 1 be an integer. If among every ( m+2 2 ) members of F all but at most m− 1 members have a common point, then there is a common point for all but at most m − 1 members of F . The claim is an extension of Helly’s theorem (m = 1). The case m = 2 was verified by Nadler and by Perles. Here we show that Eckhoff ’s conjecture follows from an old conjecture due to Szemerédi and Petruska concerning 3-uniform hypergraphs. This conjecture is still open in general; its solution for a few special cases answers Eckhoff’s problem for m = 3, 4. A new proof for the case m = 2 is also presented. Mathematics Subject Classifications: 52A10, 52A35, 05C62, 05D05, 05D15, 05C65


Introduction
The subject of this note is a combinatorial version of the classical Hadwiger-Debrunner (p, q)-problems proposed by Eckhoff [2] (see also [1]). A family F of convex sets in the plane has the ∆(m)-property if F has at least |F|−m+1 sets with non-empty intersection. We restate Eckhoff's conjecture using this notation. [2,Problem 6]) Let m 1, k = m+2 2 be integers, and let F be a family of at least k convex sets in R 2 . If every k members of F has the ∆(m)-property, then F also has the ∆(m)-property.

Problem 1. (Eckhoff
Due to Helly's theorem [5], Problem 1 has a positive answer for m = 1. The claim was verified also for m = 2 by Nadler [8] and by Perles [9]. In this note we show that Eckhoff's conjecture follows from an old conjecture due to Szemerédi and Petruska [10] on 3-uniform hypergraphs. In Section 2, Problem 1 is restated first (Problem 2) in terms of 2-representable 3uniform hypergraphs. The Szemerédi-Petruska conjecture, as reformulated by Lehel and Tuza [11,Problem 18.(a)] states that m+2 2 is the maximum order of a 3-uniform τ -critical hypergraph with transversal number m. Thus Eckhoff's conjecture becomes equivalent to a particular instance of a general extremal hypergraph problem (Theorem 6). The Szemerédi-Petruska conjecture is verified for m = 2, 3, 4 (see [7]) using the concept of 3uniform τ -critical hypergraphs, cross-intersecting set-pair systems, and τ -critical graphs; this solves Eckhoff's problem for m = 3, 4, with a new proof for m = 2 (Corollary 7).
Eckhoff made the remark that the value of k in Problem 1 is not expected to be tight. Examples in Section 5 show that k = m+2 2 cannot be lowered for m = 2, 3, but it is not optimal for m = 4. A 3-uniform hypergraph H, that is the intersection hypergraph of some family F of planar convex sets is called a 2-representable or convex hypergraph. Observe that a k-clique N ⊂ V of the intersection hypergraph indicates that the k convex sets of F corresponding to the vertices of N have a common point in the plane, due to Helly's theorem. Eckhoff's problem is stated next in terms of convex hypergraphs. Observe that by Helly's theorem, a family F of k convex sets in R 2 has the ∆(m)property if and only if the 3-uniform intersection hypergraph H defined by F has clique number ω(H) k − m + 1. This implies the equivalence of Problem 1 and Problem 2.

τ -critical 3-uniform hypergraphs
on vertex set X with all those edges in E that are contained by X. For e ∈ E, denote H −e the partial hypergraph with vertex set V and edge set E \{e}. Let H = (V, E) be the r-uniform hypergraph obtained as the complement of H with E containing all r-element subsets of V not in E.
The transversal number of a hypergraph H is defined by Let v max (r, t) be the maximum order of an r-uniform τ -critical hypergraph H with τ (H) = t. This function was introduced and investigated by Gyárfás et al. [4] and by Tuza [11,Section 4.2].
Denote ω(H) the clique number of H defined as the maximum cardinality of a subset N ⊂ V such that every r-element set of N belongs to E.  (b) Because the maximum cliques in H have no common vertex, the union of their complement in V , that is, the union of the t-element transversals of H, is equal to V . Let H be a τ -critical partial hypergraph of H with vertex V and τ (H ) = t. We claim that |V | = |V |.
Because every vertex x ∈ V \ V belongs to some t-element transversal T of H, the set T \ {x} is a (t − 1)-element transversal for all edges of H not containing x; hence τ (H ) < t, a contradiction. Thus |V | = |V | v max (r, t) follows.   As an immediate corollary of Theorem 6 and Proposition 5 we obtain an extensions of Helly's theorem together with a combinatorial proof for the case m = 2 (verified earlier by Nadler [8] and by Perles [9]).
, and let F be a family of at least k convex sets in R 2 . If every k members of F has the ∆(m)-property, then F also has the ∆(m)property.
5 Concluding remarks 5.1 The best known general bound v max (3, m) 3 4 m 2 + m + 1 is obtained by Tuza 1 using the machinery of τ -critical hypergraphs. This bound combined with Theorem 6 yields the following finiteness result on Eckhoff's problem, for every m.
Corollary 8. Let F be a family of at least k 3 4 m 2 + m + 1 convex sets in R 2 . If every k members of F has the ∆(m)-property, then F also has the ∆(m)-property.

5.2
In Corollary 7 the value of k is optimal (the smallest possible) if there is a family of n k convex sets in R 2 such that every k − 1 members of F satisfy the ∆(m)-property, but F fails it. It was proved by Nadler [8] that k = m+2 2 is optimal for m = 2, but as noted by Eckhoff [1], it is 'somewhat unlikely' that it is optimal for every m. We address optimality for m = 2, 3, 4 by defining a family F m of convex sets as follows.   Take the eight hexagons determined by the vertex sets S \ {p i , p i+1 }, 0 i 7, and take the six quadrangles Q i = (p i , p i+1 , p i+2 , p i+3 ), for i ∈ {1, 2, 3, 5, 6, 7}, with (mod 8) index arithmetic. Notice that the undefined Q 0 , Q 4 do not belong to F 4 , furthermore, the six quadrangles defined in F 4 form three disjoint pairs. Taking one quadrangle from each pair plus the eight hexagons form a subfamily of 11 convex sets with no common point, thus at most 10 = n − m members of F 4 can be covered by one point. On the other hand, three intersecting quadrangles plus seven more hexagons contained in every subfamily F 4 \ {C}, that is, (k − 1) − (m − 1) = 10 members have a common point q of the plane as it is seen in Fig.1.
Family F m shows that k = m+2 2 is optimal in Corollary 7 for m = 2, 3. Each of F 2 and F 3 is derived from a 3-uniform hypergraph witnessing v max (3, m) = m+2 2 . For m = 4 the 3-uniform witness hypergraphs are not 2-representable. This fact was observed by Jobson et al. [6] when a similar method using convex hypergraphs was applied to  such that Theorem 6 remains true when v max (3, m) is replaced with k(m). The exact values, which we know are k(1) = 3, k(2) = 6, k(3) = 10, k(4) = 14, and we ask the question whether k(m) = Ω(m 2 ).