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Families of finite sets in which no set is covered by the union of r others
, 1985
"... Let f,(n, k) denote the maximum number of ksubsets of an nset satisfying the condition in the title. It is proved that f,(n,r(t1)+1+d)(n t dl/ lk dl u t for n sufficiently large whenever d = 0,1 or d < r/2 t 2 withJh equality holding iff there exists a Steiner system S(t, r(t 1) + l, n d). ..."
Abstract

Cited by 130 (2 self)
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Let f,(n, k) denote the maximum number of ksubsets of an nset satisfying the condition in the title. It is proved that f,(n,r(t1)+1+d)(n t dl/ lk dl u t for n sufficiently large whenever d = 0,1 or d < r/2 t 2 withJh equality holding iff there exists a Steiner system S(t, r(t 1) + l, n d). The determination of f,(n, 2r) led us to a new generalization of BIRD (Definition 2.4). Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family.
A Switching Lemma for Small Restrictions and Lower Bounds for kDNF Resolution (Extended Abstract)
 SIAM J. Comput
, 2002
"... We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with kDNFs instead of cla ..."
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Cited by 46 (7 self)
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We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with kDNFs instead of clauses. We also obtain an exponential separation between depth d circuits of k + 1.
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.