Subsets of products of finite sets of positive upper density

Abstract

In this note we prove that for every sequence (mq)q of positive integers and for every real 0<δ≤1 there is a sequence (nq)q of positive integers such that for every sequence (Hq)q of finite sets such that |H q|=n q for every q∈N and for every Dkq=0k-1Hq with the property thatlimsupk|D∩q=0k-1Hq||q=0k-1Hq|δ there is a sequence (Jq)q, where J qH q and |J q|=m q for all q, such that q=0k-1JqD 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 in terms of the sequence of (mq)q.