Uniform Edge Distribution in Hypergraphs is Hereditary

Let α ∈ (0, 1), l ≥ 2 and let Hn be an l-graph on n vertices. Hn is (α, ξ)-uniform if every ξn vertices of Hn span (α± ξ) ( ξn l ) edges. Our main result is the following. For all δ̃, there exist δ, r, n0 such that, if n > n0 and H n is (α, δ)-uniform, then all but exp{−r/20} ( n r ) r-sets of vertices induce a subhypergraph that is (α, δ̃)-uniform. We also present the following application. Let F be a fixed l-graph, and c > 0. Then there is an n0 and r′ such that: If H is an n vertex l-graph (n > n0) such that the deletion of any cn edges of H leaves an l-graph that admits no homomorphism into F , then there exists H′ ⊂ H on r′ vertices, that also admits no homomorphism into F . This extends a recent result of Alon and Shapira [3], who proved it when F is a complete graph.