Measurable events indexed by words

For every integer k > 2 let [k] 1 is a subset of [k] 2, every 0 < ε 6 1 and every integer n > 1 there exists a strictly positive constant θ(k, ε, n) with the following property. If m is a given positive integer, then there exists an integer Cor(k, ε,m) such that for every Carlson–Simpson tree T of [k] ε for every t ∈ T , there exists a Carlson–Simpson tree S of dimension m with S ⊆ T and such that for every nonempty F ⊆ S we have μ ( ⋂ t∈F At ) > θ(k, ε, |F |). The proof is based, among others, on the density version of the Carlson– Simpson Theorem established recently by the authors, as well as, on a partition result – of independent interest – closely related to the work of T. J. Carlson, and H. Furstenberg and Y. Katznelson. The argument is effective and yields explicit lower bounds for the constants θ(k, ε, n).