Results 1  10
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10
Some Structural Properties of Low Rank Matrices Related to Computational Complexity
, 1997
"... We consider the conjecture stating that a matrix with rank o(n) and ones on the main diagonal must contain nonzero entries on a 2 \Theta 2 submatrix with one entry on the main diagonal. ..."
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Cited by 19 (3 self)
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We consider the conjecture stating that a matrix with rank o(n) and ones on the main diagonal must contain nonzero entries on a 2 \Theta 2 submatrix with one entry on the main diagonal.
SetSystems with Restricted Multiple Intersections and Explicit Ramsey Hypergraphs
, 2001
"... We give a generalization for the DezaFranklSinghi Theorem in case of multiple intersections. More exactly, we prove, that if H is a setsystem, which satisfies that for some k, the kwise intersections occupy only ` residueclasses modulo a p prime, while the sizes of the members of H are not in t ..."
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Cited by 7 (1 self)
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We give a generalization for the DezaFranklSinghi Theorem in case of multiple intersections. More exactly, we prove, that if H is a setsystem, which satisfies that for some k, the kwise intersections occupy only ` residueclasses modulo a p prime, while the sizes of the members of H are not in these residue classes, then the size of H is at most (k \Gamma 1) ` X i=0 / n i ! This result considerably strengthens an upper bound of Furedi (1983), and gives partial answer to a question of T. S'os (1976). As an application, we give explicit constructions for coloring the ksubsets of an n element set with t colors, such that no monochromatic complete hypergraph on exp(c k (log n log log n) 1=t ) vertices exists. By our best knowledge, this is the first explicit construction of a Ramseyhypergraph.
kwise SetIntersections and kwise HammingDistances
, 2001
"... We prove a version of the RayChaudhuriWilson and FranklWilson theorems for k wise intersections and also generalize a classical codetheoretic result of Delsarte for kwise Hamming distances. A set of codewords a 1 ; a 2 ; : : : ; a k of length n have kwise Hammingdistance `, if there ..."
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Cited by 7 (2 self)
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We prove a version of the RayChaudhuriWilson and FranklWilson theorems for k wise intersections and also generalize a classical codetheoretic result of Delsarte for kwise Hamming distances. A set of codewords a 1 ; a 2 ; : : : ; a k of length n have kwise Hammingdistance `, if there are exactly ` such coordinates, where not all of their coordinates coincide (alternatively, exactly n \Gamma ` of their coordinates are the same). We show a Delsartelike upper bound: codes with few kwise Hammingdistances must contain few codewords. 1
Exponents of Uniform LSystems
 J. Combin. Theory (A
, 1995
"... We have determined all exponents of (n; k; L)systems of k 12 except for essentially two cases, which are related to the Steiner systems S(11; 5; 4) and S(12; 6; 5). This requires several new constructions. Also some refinements of the previous methods are necessary to get suitable upper bounds. 1 ..."
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Cited by 5 (2 self)
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We have determined all exponents of (n; k; L)systems of k 12 except for essentially two cases, which are related to the Steiner systems S(11; 5; 4) and S(12; 6; 5). This requires several new constructions. Also some refinements of the previous methods are necessary to get suitable upper bounds. 1
On kwise SetIntersections and kwise Hamming Distances
 J. COMBIN. THEORY SER. A
, 2001
"... We prove a version of the RayChaudhuriWilson and FranklWilson theorems for k wise intersections and also generalize a classical codetheoretic result of Delsarte for kwise Hamming distances. A set of codewords a 1 ; a 2 ; : : : ; a k of length n have kwise Hammingdistance `, if ther ..."
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Cited by 1 (0 self)
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We prove a version of the RayChaudhuriWilson and FranklWilson theorems for k wise intersections and also generalize a classical codetheoretic result of Delsarte for kwise Hamming distances. A set of codewords a 1 ; a 2 ; : : : ; a k of length n have kwise Hammingdistance `, if there are exactly ` such coordinates, where not all of their coordinates coincide (alternatively, exactly n ` of their coordinates are the same). We show a Delsartelike upper bound: codes with few kwise Hammingdistances must contain few codewords.
On a conjecture of Thomassen
, 2009
"... In 1983 C. Thomassen conjectured that for every k, g ∈ N there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. Kühn and Osthus (2004) proved the case g = 6. We give another proof for the case g = 6 which is based on ..."
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In 1983 C. Thomassen conjectured that for every k, g ∈ N there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. Kühn and Osthus (2004) proved the case g = 6. We give another proof for the case g = 6 which is based on a result of Füredi (1983) about hypergraphs. We also show that the analogous conjecture for directed graphs is true.
A survey of Turán problems for expansions
, 2015
"... The rexpansion G+ of a graph G is the runiform hypergraph obtained from G by enlarging each edge of G with a vertex subset of size r − 2 disjoint from V (G) such that distinct edges are enlarged by disjoint subsets. Let exr(n, F) denote the maximum number of edges in an runiform hypergraph with n ..."
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The rexpansion G+ of a graph G is the runiform hypergraph obtained from G by enlarging each edge of G with a vertex subset of size r − 2 disjoint from V (G) such that distinct edges are enlarged by disjoint subsets. Let exr(n, F) denote the maximum number of edges in an runiform hypergraph with n vertices not containing any copy of the runiform hypergraph F. Many problems in extremal set theory ask for the determination of exr(n,G +) for various graphs G. We survey these Turántype problems, focusing on recent developments. 1
AlmostFisher families
"... A classic theorem in combinatorial design theory is Fisher’s inequality, which states that a family F of subsets of [n] with all pairwise intersections of size λ can have at most n nonempty sets. One may weaken the condition by requiring that for every set in F, all but at most k of its pairwise in ..."
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A classic theorem in combinatorial design theory is Fisher’s inequality, which states that a family F of subsets of [n] with all pairwise intersections of size λ can have at most n nonempty sets. One may weaken the condition by requiring that for every set in F, all but at most k of its pairwise intersections have size λ. We call such families kalmost λFisher. Vu was the first to study the maximum size of such families, proving that for k = 1 the largest family has 2n − 2 sets, and characterising when equality is attained. We substantially refine his result, showing how the size of the maximum family depends on λ. In particular we prove that for small λ one essentially recovers Fisher’s bound. We also solve the next open case of k = 2 and obtain the first nontrivial upper bound for general k. 1