Weak hypergraph regularity and linear hypergraphs

Abstract

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers ℓ ⩾ k ⩾ 2 and every d > 0 there exists ϱ > 0 for which the following holds: if H is a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size ϱn is at least d, then H contains every linear k-uniform hypergraph F with ℓ vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph ε-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems.