On -partite hypergraphs with the induced -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 k -partite k -uniform hypergraph G we say that a k -partite k -uniform hypergraph R = ( W 1 , … , W k , F ) has the induced ε -density property if every subhypergraph of R 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 = ( V 1 , … , V k , E ) with | V 1 | , … , | V k | ≤ n . We give several proofs of this result, some of which allow for the hypergraph R to be taken with at most 2 2 c n k − 1 vertices.