Results 1 
9 of
9
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 159 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 44 (7 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
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 hypergraph Turán theorem via lagrangians of intersecting families, submitted
"... Let K33,3 be the 3graph with 15 vertices {xi, yi: 1 ≤ i ≤ 3} and {zij: 1 ≤ i, j ≤ 3}, and 11 edges {x1, x2, x3}, {y1, y2, y3} and {{xi, yj, zij} : 1 ≤ i, j ≤ 3}. We show that for large n, the unique largest K33,3free 3graph on n vertices is a balanced blowup of the complete 3graph on 5 vertices ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Let K33,3 be the 3graph with 15 vertices {xi, yi: 1 ≤ i ≤ 3} and {zij: 1 ≤ i, j ≤ 3}, and 11 edges {x1, x2, x3}, {y1, y2, y3} and {{xi, yj, zij} : 1 ≤ i, j ≤ 3}. We show that for large n, the unique largest K33,3free 3graph on n vertices is a balanced blowup of the complete 3graph on 5 vertices. Our proof uses the stability method and a result on lagrangians of intersecting families that has independent interest. 1
Rainbow numbers for small stars with one edge added
 DISCUSSIONES MATHEMATICAE GRAPH THEORY
, 2010
"... A subgraph of an edgecolored graph is rainbow if all of its edges have dierent colors. For a graph H and a positive integer n, the antiRamsey number f(n; H) is the maximum number of colors in an edgecoloring of Kn with no rainbow copy of H. The rainbow number rb(n; H) is the minimum number of col ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
A subgraph of an edgecolored graph is rainbow if all of its edges have dierent colors. For a graph H and a positive integer n, the antiRamsey number f(n; H) is the maximum number of colors in an edgecoloring of Kn with no rainbow copy of H. The rainbow number rb(n; H) is the minimum number of colors such that any edgecoloring of Kn with rb(n; H) number of colors contains a rainbow copy of H. Certainly rb(n; H) = f(n; H) + 1. AntiRamsey numbers were introduced by Erdos et al. [5] and studied in numerous papers. We show that rb(n; K1;4 + e) = n + 2 in all nontrivial cases.