On k-partite hypergraphs with the induced ε-density property

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