A density version of the Carlson--Simpson theorem

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 $$\limsup_{n\to\infty} \frac{|A\cap [k]^n|}{k^n}>0$$ 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.Comment: 73 pages, no figure

Partition theorems for left and right variable words

In 1984 T. Carlson and S. Simpson established an infinitary extension of the Hales-Jewett Theorem in which the leftmost letters of all but one of the words were required to be variables. (We call such words left variable words.) In this paper we extend the Carlson-Simpson result for left variable words, prove a corresponding result about right variable words, and determine precisely the extent to which left and right variable words can be combined in such extensions. The results mentioned so far all involve a finite alphabet. We show that the results for left variable words do not extend to words over an infinite alphabet, but that the results for right variable words do extend

this is the final version as it was submitted to the publisher. – NH Partition Theorems for Left and Right Variable Words

Abstract. In 1984 T. Carlson and S. Simpson established an infinitary extension of the Hales-Jewett Theorem in which the leftmost letters of all but one of the words were required to be variables. (We call such words left variable words.) In this paper we extend the Carlson-Simpson result for left variable words, prove a corresponding result about right variable words, and determine precisely the extent to which left and right variable words can be combined in such extensions. The results mentioned so far all involve a finite alphabet. We show that the results for left variable words do not extend to words over an infinite alphabet, but that the results for right variable words do extend. 1