Subsets of products of finite sets of positive upper density

In this note we prove that for every sequence (mq)q(mq)q of positive integers and for every real 0<δ≤10<δ≤1 there is a sequence (nq)q(nq)q of positive integers such that for every sequence (Hq)q(Hq)q of finite sets such that |Hq|=nq|Hq|=nq for every q∈Nq∈N and for every D⊆⋃k∏q=0k−1Hq with the property thatlimsupk|D∩∏q=0k−1Hq||∏q=0k−1Hq|≥δ there is a sequence (Jq)q(Jq)q, where Jq⊆HqJq⊆Hq and |Jq|=mq|Jq|=mq for all q  , such that ∏q=0k−1Jq⊆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 (nq)q(nq)q in terms of the sequence of (mq)q(mq)q.