A density version of the Carlson–Simpson theorem

Pandelis Dodos; Vassilis Kanellopoulos; Konstantinos Tyros

doi:10.4171/jems/484

JournalsjemsVol. 16, No. 10pp. 2097–2164

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

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Abstract

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

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

n \to \infty lim sup ∣ 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 {c^{⌢} w_{0} (a_{0})^{⌢} .. .^{⌢} w_{n} (a_{n}) : n \in N and a_{0}, ..., a_{n} \in [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