# A density version of the Carlson–Simpson theorem

### Pandelis Dodos

University of Athens, Greece### Vassilis Kanellopoulos

National Technical University of Athens, Greece### Konstantinos Tyros

University of Toronto, Canada

## 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*

$n→∞limsup ∣A∩[k]_{n}∣/k_{n}>0$

*there exist a word $c$ over $k$ and a sequence $(w_{n})$ of left variable words over $k$ such that the set*

${c}∪{c_{⌢}w_{0}(a_{0})_{⌢}..._{⌢}w_{n}(a_{n}):n∈Nanda_{0},...,a_{n}∈[k]}$

*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.

## Cite this article

Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros, A density version of the Carlson–Simpson theorem. J. Eur. Math. Soc. 16 (2014), no. 10, pp. 2097–2164

DOI 10.4171/JEMS/484