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 $0<\delta ⩽1$ there is a sequence $(n_q{)}_q$ of positive integers such that for every sequence $(H_q{)}_q$ of finite sets such that $\mathrm{\mid }{H}_q\mathrm{\mid }=n_q$ for every $q\in \mathbb{N}$ and for every $D\subseteq {\bigcup }_k{\prod }_{q=0}^{k-1}{H}_q$ with the property that $$\underset{k}{\mathrm{lim\hspace{0.17em}sup}}\frac{\mathrm{\mid }D\cap \prod _{q=0}^{k-1}{H}_q\mathrm{\mid }}{\mathrm{\mid }\prod _{q=0}^{k-1}{H}_q\mathrm{\mid }}⩾\delta$$ there is a sequence $(J_q{)}_q$, where $J_q\subseteq H_q$ and $\mathrm{\mid }{J}_q\mathrm{\mid }=m_q$ for all $q$, such that ${\prod }_{q=0}^{k-1}{J}_q\subseteq D$ for infinitely many $k\mathrm{.}$ 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 $(m_q{)}_q$.