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Large cliques or cocliques in hypergraphs with forbidden order-size pairs

Abstract

The well-known Erdős-Hajnal conjecture states that for any graph FF, there exists ϵ>0ϵ>0 such that every nn-vertex graph GG that contains no induced copy of FF has a homogeneous set of size at least nϵn^ϵ. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on mm vertices and ff edges for any positive mm and 0f(m2)0≤f≤(m2), then we obtain large homogeneous sets. For triple systems, in the first nontrivial case m=4m=4, for every S0,1,2,3,4S⊆{0,1,2,3,4}, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in SS. In most cases the bounds are essentially tight. We also determine, for all SS, whether the growth rate is polynomial or polylogarithmic. Some open problems remain

Similar works

This paper was published in KITopen.

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