Introduction Helly's theorem asserts that if F is a finite family of convex sets in R d in which every d + 1 or fewer sets have a point in common then T F 6= ;. Our starting point, the (p; q) theorem, is a deep extension of Helly's theorem. It was conjectured by Hadwiger and Debrunner and proved by Alon and Kleitman [3]. Let p q 2 be integers. A family F of convex sets in R d is said to have the (p; q) property if among every p sets of F , some q have a point in common. Theorem 1 ((p; q) theorem, Alon & Kleitmen) For every p q d+ 1 there exists a number C =

Let A be a family of convex sets in R d . A line transversal to A is a line which intersects every member of A. More generally, a k-tranversal to A is an ane subspace of dimension k which intersects every member of A.

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.

A set of the form C " Z d , where C ` R d is convex and Z d denotes the integer lattice, is called a convex lattice set. It is known that the Helly number of d-dimensional convex lattice sets is 2 d . We prove that the fractional Helly number is only d + 1: for every d and every ff 2 (0; 1] there exists fi ? 0 such that whenever F 1 ; : : : ; Fn are convex lattice sets in Z d such that T i2I F i 6= ; for at least ff \Gamma n d+1 \Delta index sets I ` f1; 2; : : : ; ng of size d + 1 then there exists a (lattice) point common to at least fin of the F i . This implies a (p; d + 1)-theorem for every p d + 1; that is, if F is a finite family of convex lattice sets in Z d such that among every p sets of F , some d + 1 intersect, then F has a transversal of size bounded by a function of d and p. 1

Let P 1 ; P 2 ; : : : ; P d+1 be pairwise disjoint n-element point sets in general position in d-space. It is shown that there exist a point O and suitable subsets Q i P i (i = 1; 2; : : : ; d + 1) such that jQ i j c d jP i j, and every d-dimensional simplex with exactly one vertex in each Q i contains O in its interior. Here c d is a positive constant depending only on d.

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Afamilyofsetshasthe(p, q) property if among any p members of the family some q have a nonempty intersection. The authors have proved that for every p # q # d +1 there is a c = c(p, q, d) < # such that for every family F of compact, convex sets in R which has the (p, q) property there is a set of at most c points in R that intersects each member of F , thus settling an old problem of Hadwiger and Debrunner. Here we present a purely combinatorial proof of this result.

We prove an upper bound, tight up to a factor of 2, for the number of vertices of level at most ℓ in an arrangement of n halfspaces in R d, for arbitrary n and d (in particular, the dimension d is not considered constant). This partially settles a conjecture of Eckhoff, Linhart, and Welzl. Up to the factor of 2, the result generalizes McMullen’s Upper Bound Theorem for convex polytopes (the case ℓ = 0) and extends a theorem of Linhart for the case d ≤ 4. Moreover, the bound sharpens asymptotic estimates obtained by Clarkson and Shor. The proof is based on the h-matrix of the arrangement (a generalization, introduced by Mulmuley, of the h-vector of a convex polytope). We show that bounding appropriate sums of entries of this matrix reduces to a lemma about quadrupels of sets with certain intersection properties, and we prove this lemma, up to a factor of 2, using tools from multilinear algebra. This extends an approach of Alon and Kalai, who used linear algebra methods for an alternative proof of the classical Upper Bound Theorem. The bounds for the entries of the h-matrix also imply bounds for the number of i-dimensional faces, i> 0, at level at most ℓ. Furthermore, we discuss a connection with crossing numbers of graphs that was one of the main motivations for investigating exact bounds that are valid for arbitrary dimensions. 1.

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Radon's Theorem: If A is a set of d + 2 points in R d , then there are disjoint subsets A 1 ; A 2 ae A such that conv(A 1 ) " conv(A 2 ) 6= ;. Carath'eodory's Theorem: If X ` R d and x 2 conv(X) then x 2 conv(Y ) for some at most (d + 1)-point Y ` X. Ham-Sandwich Cut Theorem: Any d finite sets in R d can be simultaneously bisected by a hyperplane. A hyperplane h bisects a finite set A if each of the open halfspaces defined by h contains at most