Subsets of products of finite sets of positive upper density

In this note we prove that for every sequence ( m q ) q of positive integers and for every real 0 < δ ≤ 1 there is a sequence ( n q ) q of positive integers such that for every sequence ( H q ) q of finite sets such that | H q | = n q for every q ∈ N and for every D ⊆ ⋃ k ∏ q = 0 k − 1 H q with the property that l i m s u p k | D ∩ ∏ q = 0 k − 1 H q | | ∏ q = 0 k − 1 H q | ≥ δ there is a sequence ( J q ) q , where J q ⊆ H q and | J q | = m q for all q, such that ∏ q = 0 k − 1 J q ⊆ D for infinitely many k. This gives us a density version of a well-known Ramsey-theoretic result. We also give some estimates on the sequence ( n q ) q in terms of the sequence of ( m q ) q .