Erdős-Hajnal-type theorems in hypergraphs

Abstract

The Erdős–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^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(log n)^½ + δ(ℋ), where δ(ℋ) > 0 depends only on ℋ. This improves on the bound of c(log n)^½, 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 Erdős–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.