On kk-partite hypergraphs with the induced εε-density property

In this paper we extend the study of bipartite graphs with the induced εε-density property introduced by Frankl, Rödl, and the author. For a given kk-partite kk-uniform hypergraph GG we say that a kk-partite kk-uniform hypergraph R=(W1,…,Wk,F)R=(W1,…,Wk,F) has the induced  εε-density property   if every subhypergraph of RR with at least ε|F|ε|F| edges contains a copy of GG which is an induced subhypergraph of RR. We show that for every ε>0ε>0 and positive integers kk and nn there exists a kk-partite kk-uniform hypergraph RR with the induced εε-density property for every G=(V1,…,Vk,E)G=(V1,…,Vk,E) with |V1|,…,|Vk|≤n|V1|,…,|Vk|≤n. We give several proofs of this result, some of which allow for the hypergraph RR to be taken with at most 22cnk−122cnk−1 vertices.