Erdos–Hajnal-type theorems in hypergraphs

Abstract

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[superscript δ(H)], where δ(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)[superscript 1/2 + δ(H)], where δ(H) > 0 depends only on H. This improves on the bound of c(logn)[superscript 1/]2, which holds in all 3-uniform hypergraphs, and, up to the value of the constant δ(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.