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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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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.
My joint work with Richard Rado
 In Proc. 11th British Combinat. Conf., Cambr
, 1987
"... I first became aware of Richard Rado's existence in 1933 when his important paper Studien zur Kombinatorik appeared. I thought a great deal about the many fascinating and deep unsolved problems stated in this paper but I never succeeded to obtain any significant results here and since I have to repo ..."
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I first became aware of Richard Rado's existence in 1933 when his important paper Studien zur Kombinatorik appeared. I thought a great deal about the many fascinating and deep unsolved problems stated in this paper but I never succeeded to obtain any significant results here and since I have to report here about our joint work I will mostly ignore these questions. Our joint work extends to more than 50 years; we wrote 18 joint papers, several of them jointly with A. Hajnal, three with E. Milner, one with F. Galvin, one with Chao Ko, and we have a book on partition calculus with A. Hajnal and A. Máté. Our most important work is undoubtedly in set theory and, in particular, the creation of the partition calculus. The term partition calculus is, of course, due to Rado. Without him, I often would have been content in stating only special cases. We started this work in earnest in 1950 when I was at University College and Richard in King's College. We completed a fairly systematic study of this subject in 1956, but soon after this we started to collaborate with A. Hajnal, and by 1965 we published our GTP (Giant Triple Paper this terminology was invented by Hajnal) which, I hope, will outlive the authors by a long time. I would like to write by centuries if the reader does not
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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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.
Traces Without Maximal Chains
"... The trace of a family of sets A on a set X is AX = {A ∩ X: A ∈ A}. If A is a family of ksets from an nset such that for any rsubset X the trace AX does not contain a maximal chain, then how large can A be? Patkós conjectured that, for n sufficiently large, the size of A is at most ( n−k+r−1) r− ..."
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The trace of a family of sets A on a set X is AX = {A ∩ X: A ∈ A}. If A is a family of ksets from an nset such that for any rsubset X the trace AX does not contain a maximal chain, then how large can A be? Patkós conjectured that, for n sufficiently large, the size of A is at most ( n−k+r−1) r−1. Our aim in this paper is to prove this conjecture. 1
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
C0UNTERE~AMPLE TO TiiE FRANKLPACH CONJECTURE FOR UNIFORM, DENSE FAMILIES
, 1996
"... N denotes the set of positive integers and for e, nEN, e