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Transversal Numbers for Hypergraphs Arising in Geometry
 Adv. Appl. Math
, 2001
"... 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 ..."
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Cited by 16 (3 self)
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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 =
Progress in Geometric Transversal Theory
 Advances in Discrete and Computational Geometry
, 2001
"... 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 ktranversal to A is an ane subspace of dimension k which intersects every member of A. ..."
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Cited by 13 (2 self)
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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 ktranversal to A is an ane subspace of dimension k which intersects every member of A.
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 11 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension 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 Fractional Helly theorem for convex lattice sets
, 2001
"... 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 ddimensional convex lattice sets is 2 d . We prove that the fractional Helly number is only d + 1: for every d and every ff 2 ..."
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Cited by 7 (1 self)
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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 ddimensional 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
On a geometric generalization of the upper bound theorem
 In FOCS: IEEE Symposium on Foundations of Computer Science (FOCS
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A Helly Type Theorem For Hypersurfaces
, 1987
"... It is proved that if H,,..., H, are hypersurfaces of degree at most d in ndimensional projective space and P,,..., P, are points so that Pi 4 Hi for all i and P, E H, for 1 < i < j < m then m < ( ” 2 d). ..."
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Cited by 3 (0 self)
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It is proved that if H,,..., H, are hypersurfaces of degree at most d in ndimensional projective space and P,,..., P, are points so that Pi 4 Hi for all i and P, E H, for 1 < i < j < m then m < ( ” 2 d).
A Purely Combinatorial Proof of the Hadwiger Debrunner (p, q) Conjecture
"... 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 s ..."
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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.