Two Erdős–Hajnal-type theorems in hypergraphs

The Erdős–Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs. A theorem of Rödl asserts that if an n-vertex graph is non-universal then it contains an almost homogeneous set (i.e. one with edge density either very close to 0 or 1) of size Ω(n). We prove that if a 3-uniform hypergraph is non-universal then it contains an almost homogeneous set of size Ω(log⁡n). An example of Rödl from 1986 shows that this bound is tight. Let R_r(t) denote the size of the largest non-universal r-graph G so that neither G nor its complement contain a complete r-partite subgraph with parts of size t. We prove an Erdős–Hajnal-type stepping-up lemma, showing how to transform a lower bound for R_r(t) into a lower bound for R_r+1(t). As an application of this lemma, we improve a bound of Conlon–Fox–Sudakov by showing that R₃(t)≥t^Ω(t).