A density version of the Carlson–Simpson theorem

Abstract

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following.

For every integer $k ≥ 2$ and every set $A$ of words over $k$ satisfying

$\mathrm {lim \ sup}_{n\to\infty} |A \cap [k]^n|/k^n >0$

there exist a word $c$ over $k$ and a sequence $(w_n)$ of left variable words over $k$ such that the set

${c}\cup \big\{c^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_n(a_n):n\in\mathbb N \text{ and } a_0,...,a_n\in [k]\big\}$

is contained in $A$.

While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.