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Crossing numbers and hard Erdős problems in discrete geometry
 COMBINATORICS, PROBABILITY AND COMPUTING
, 1997
"... We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the min ..."
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We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.
On the VCdimension of uniform hypergraphs
 Journal of Algebraic Combinatorics
"... Let F be a kuniform hypergraph on [n] where k − 1 is a power of some prime p and n ≥ n0(k). Our main result says that if F > () n k−1 − logp n + k!kk, then there exists E0 ∈ F such that {E ∩ E0) : E ∈ F} contains all subsets of E0. This improves a due to Frankl and Pach [7]. longstanding bound of ..."
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Cited by 3 (0 self)
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Let F be a kuniform hypergraph on [n] where k − 1 is a power of some prime p and n ≥ n0(k). Our main result says that if F > () n k−1 − logp n + k!kk, then there exists E0 ∈ F such that {E ∩ E0) : E ∈ F} contains all subsets of E0. This improves a due to Frankl and Pach [7]. longstanding bound of ( n k−1 1.
Forbidding complete hypergraphs as traces
, 2006
"... Let 2 ≤ q ≤ min{p, t − 1} be fixed and n → ∞. Suppose that F is a puniform hypergraph on n vertices that contains no complete quniform hypergraph on t vertices as a trace. We determine the asymptotic maximum size of F in many cases. For example, when q = 2 and p ∈ {t, t + 1}, the maximum is ( n t− ..."
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Let 2 ≤ q ≤ min{p, t − 1} be fixed and n → ∞. Suppose that F is a puniform hypergraph on n vertices that contains no complete quniform hypergraph on t vertices as a trace. We determine the asymptotic maximum size of F in many cases. For example, when q = 2 and p ∈ {t, t + 1}, the maximum is ( n t−1)t−1 + o(n t−1), and when p = t = 3, it is ⌊ (n−1)2 4 ⌋ for all n ≥ 3. Our proofs use the KruskalKatona theorem, an extension of the sunflower lemma due to Füredi, and recent results on hypergraph Turán numbers.
Set Systems and Families of Permutations with Small Traces
"... We study the maximum size of a set system on n elements whose trace on any b elements has size at most k. This question extends to hypergraphs the classical Diractype problems from extremal graph theory. We show that if for some b ≥ i ≥ 0 the shatter function fR of a set system ([n], R) satisfies f ..."
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We study the maximum size of a set system on n elements whose trace on any b elements has size at most k. This question extends to hypergraphs the classical Diractype problems from extremal graph theory. We show that if for some b ≥ i ≥ 0 the shatter function fR of a set system ([n], R) satisfies fR(b) < 2 i (b − i + 1) then R  = O(n i); this generalizes Sauer’s Lemma on the size of set systems with bounded VCdimension. We use this bound to delineate the main growth rates for the same problem on families of permutations, where the trace corresponds to the inclusion for permutations. This is related to a question of Raz on families of permutations with bounded VCdimension that generalizes the StanleyWilf conjecture on permutations with excluded patterns. 1
Minimum Convex Partitions and Maximum Empty Polytopes ∗
, 2012
"... Let S be a set of n points in R d. A Steiner convex partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a Steiner convex partition with at most ⌈(n − 1)/d ⌉ tiles. This bound is the best possible for points in general position in the plane, and it is ..."
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Let S be a set of n points in R d. A Steiner convex partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a Steiner convex partition with at most ⌈(n − 1)/d ⌉ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d ≥ 3. We also give the first constantfactor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n). Here we give a (1−ε)approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the ddimensional unit box [0,1] d.
Computational Geometry Column 53
"... This column is devoted to partitions of pointsets into convex subsets with interiordisjoint convex hulls. We review some partitioning problems and corresponding algorithms. At the end we list some open problems. For simplicity, in most cases we remain at the lowest possible interesting level, that ..."
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This column is devoted to partitions of pointsets into convex subsets with interiordisjoint convex hulls. We review some partitioning problems and corresponding algorithms. At the end we list some open problems. For simplicity, in most cases we remain at the lowest possible interesting level, that is, in the plane.
Multivalued generalizations of the Frankl–Pach Theorem
, 2011
"... In [13] P. Frankl and J. Pach proved the following uniform version of Sauer’s Lemma. Let n, d, s be natural numbers such that d ≤ n, s + 1 ≤ n/2. Let F ⊆ ([n]) d be an arbitrary duniform set system such that F does not shatter an s + 1element set, then n ..."
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In [13] P. Frankl and J. Pach proved the following uniform version of Sauer’s Lemma. Let n, d, s be natural numbers such that d ≤ n, s + 1 ≤ n/2. Let F ⊆ ([n]) d be an arbitrary duniform set system such that F does not shatter an s + 1element set, then n