Erdos-Hajnal-type theorems in hypergraphs

TL;DR

This paper extends Erdős-Hajnal-type results to 3-uniform hypergraphs, showing they contain large complete or empty tripartite subgraphs, and demonstrates that similar bounds do not hold for higher uniformities.

Contribution

It proves a hypergraph analogue of the Erdős-Hajnal theorem for 3-uniform hypergraphs and shows the conjecture fails for k > 3.

Findings

3-uniform hypergraphs contain large tripartite subgraphs of size c(log n)^{1/2 + d(H)}

Improves bounds on large homogeneous substructures in hypergraphs

Erdős-Hajnal-type bounds do not extend to k-uniform hypergraphs for k > 3

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^{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 a 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)^{1/2 + d(H)}, where d(H) > 0 depends only on H. This improves on the bound of c(log n)^{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 > 3, 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.