Two Erdős–Hajnal-type theorems in hypergraphs

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