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 lim sup n→∞ |A ∩ [k] n | k n > 0 there exist a word c over k and a sequence (wn) of left variable words over k such that the set {c} ∪ c w 0 (a 0 ) ... wn(an) : n ∈ N and a 0 , ..., an ∈ [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.2000 Mathematics Subject Classification: 05D10.

Measurable events indexed by words

“…Theorem 2.8 is the density version of a well-known coloring result due to T. J. Carlson and S. J. Simpson [4]. Also we notice that the argument in [7] is effective and gives explicit upper bounds for the numbers DCS(k, m, δ). These upper bounds, however, have an Ackermann-type dependence with respect to k.…”

“…On the proof of Theorem 1.2. The first basic ingredient of the proof of Theorem 1.2 is the density version of the Carlson-Simpson Theorem established, recently, in [7]. The second basic ingredient is a partition result closely related to the work of T. J. Carlson [3], and H. Furstenberg and Y. Katznelson [9].…”

A concentration inequality for product spaces

“…There is a natural extension of Theorem 1 which deals simultaneously with a family of random variables. Although in applications one usually encounters only finite families of random variables (see, e.g., [5]), the cleanest formulation of this extension is for stochastic processes indexed by the sample space of a probability space (T, Σ, µ). Specifically, we have the following theorem.…”

“…Theorem 1 was motivated by these results and was found in an effort to abstract their probabilistic features. We expect that Theorem 1 will in turn facilitate further applications, possibly even beyond the combinatorial context of [4,5].…”

Measurable events indexed by products of trees

“…This methodology has been applied successfully to related problems in Ramsey Theory -see, in particular, [6].…”