Discrete Applied Mathematics - Combinatorics and complexity
Non-Ramsey graphs are c log n-universal
Journal of Combinatorial Theory Series A
Hypergraphs, quasi-randomness, and conditions for regularity
Journal of Combinatorial Theory Series A
Regularity lemma for k-uniform hypergraphs
Random Structures & Algorithms
Crossing patterns of semi-algebraic sets
Journal of Combinatorial Theory Series A
Quasirandomness, Counting and Regularity for 3-Uniform Hypergraphs
Combinatorics, Probability and Computing
The counting lemma for regular k-uniform hypergraphs
Random Structures & Algorithms
On graphs with linear Ramsey numbers
Journal of Graph Theory
The Erdős--Hajnal conjecture for bull-free graphs
Journal of Combinatorial Theory Series B
Two remarks on the Burr-Erdős conjecture
European Journal of Combinatorics
Weak hypergraph regularity and linear hypergraphs
Journal of Combinatorial Theory Series B
On two problems in graph Ramsey theory
Combinatorica
An improved bound for the stepping-up lemma
Discrete Applied Mathematics
Complete Partite subgraphs in dense hypergraphs
Random Structures & Algorithms
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The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^@d^(^H^), where @d(H)0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that an H-free graph must contain a complete or empty bipartite graph with parts of polynomial size. We prove an analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform hypergraph on n vertices is H-free, for any given H, then it must contain a complete or empty tripartite subgraph with parts of order c(logn)^1^2^+^@d^(^H^), where @d(H)0 depends only on H. This improves on the bound of c(logn)^1^2, which holds in all 3-uniform hypergraphs, and, up to the value of the constant @d(H), is best possible. We also prove that, for k=4, no analogue of the standard Erdos-Hajnal conjecture can hold in k-uniform hypergraphs. That is, there are k-uniform hypergraphs H and sequences of H-free hypergraphs which do not contain cliques or independent sets of size appreciably larger than one would normally expect.