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 R 3 (t) ≥ t Ω(t) . √ log n) , that is, a significantly larger set than in the worst case. The notorious Erdős-Hajnal Conjecture states that one should be able to go even further and improve this bound to n c , where c = c(H). We refer the reader to [3] for more background on this conjecture and related results. Supported in part by ERC Starting Grant 633509.Conlon, Fox and Sudakov [6] and Rödl and Schacht [16] have recently initiated the study of problems of this type in the setting of r-uniform hypergraphs (or r-graphs for short). Our aim in this paper is to obtain two results of this flavor which are described in the next two subsections.
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