A density version of the Carlson–Simpson theorem

Dodos, Pandelis, Kanellopoulos, Vassilis, and Tyros, Konstantinos. "A density version of the Carlson–Simpson theorem." Journal of the European Mathematical Society 016.10 (2014): 2097-2164. <http://eudml.org/doc/277175>.

@article{Dodos2014,
abstract = {We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k \ge 2$ and every set $A$ of words over $k$ satisfying $\mathrm \{lim \ sup\}_\{n\rightarrow \infty \} |A \cap [k]^n|/k^n >0$ 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 \lbrace c^\{\UnimplementedOperator \}w_0(a_0)^\{\UnimplementedOperator \}...^\{\UnimplementedOperator \}w_n(a_n):n\in \mathbb \{N\} \text\{ and \} a_0,...,a_n\in [k]\big \rbrace $ 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.},
author = {Dodos, Pandelis, Kanellopoulos, Vassilis, Tyros, Konstantinos},
journal = {Journal of the European Mathematical Society},
keywords = {words; left variable words; density; words; left variable words; density},
language = {eng},
number = {10},
pages = {2097-2164},
publisher = {European Mathematical Society Publishing House},
title = {A density version of the Carlson–Simpson theorem},
url = {http://eudml.org/doc/277175},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Dodos, Pandelis
AU - Kanellopoulos, Vassilis
AU - Tyros, Konstantinos
TI - A density version of the Carlson–Simpson theorem
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 10
SP - 2097
EP - 2164
AB - We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k \ge 2$ and every set $A$ of words over $k$ satisfying $\mathrm {lim \ sup}_{n\rightarrow \infty } |A \cap [k]^n|/k^n >0$ 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 \lbrace c^{\UnimplementedOperator }w_0(a_0)^{\UnimplementedOperator }...^{\UnimplementedOperator }w_n(a_n):n\in \mathbb {N} \text{ and } a_0,...,a_n\in [k]\big \rbrace $ 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.
LA - eng
KW - words; left variable words; density; words; left variable words; density
UR - http://eudml.org/doc/277175
ER -