Subsets of Products of Finite Sets of Positive Upper Density

Abstract

In this note we prove that for every sequence $(m_q{)}_q$ of positive integers and for every real $0<\delta ⩽1$ 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 $q\in \mathbb{N}$ and for every $D\subseteq {\bigcup }_k{\prod }_{q=0}^{k-1}{H}_q$ with the property that $$\underset{k}{\mathrm{limsup}}\frac{|D\cap {\prod }_{q=0}^{k-1}{H}_q|}{|{\prod }_{q=0}^{k-1}{H}_q|}⩾\delta$$ there is a sequence $(J_q{)}_q$, where $J_q\subseteq H_q$ and $|J_q|=m_q$ for all $q$, such that ${\prod }_{q=0}^{k-1}{J}_q\subseteq D$ 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 $(m_q{)}_q$.

12 pages