Subsets of Products of Finite Sets of Positive Upper Density

TL;DR

This paper proves a density version of a classical Ramsey-theoretic result, showing that subsets with positive upper density contain structured products of finite sets, with explicit bounds on the sizes involved.

Contribution

It introduces a density-based extension of a known Ramsey theorem and provides estimates for the sizes of the finite sets involved.

Findings

Existence of sequences ensuring structured products within dense subsets

Explicit bounds on the sizes of finite sets for the construction

Extension of classical Ramsey results to density contexts

Abstract

In this note we prove that for every sequence $(m_q)_{q}$ of positive integers and for every real $0<\delta\leqslant1$ 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 $$\limsup_k \frac{|D\cap \prod_{q=0}^{k-1} H_q|}{|\prod_{q=0}^{k-1}H_q|}\geqslant\delta$$ 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 $(m_q)_{q}$.