Measurable events indexed by words

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For every integer k≥ 2 let [k]<ⁿ be the set of all words over k, that is, all finite sequences having values in [k]:={1,...,k}. A Carlson-Simpson tree of [k]<ⁿ of dimension m≥ 1 is a subset of [k]<ⁿ of the form {w}∪ {w₀(a₀)...ₙ(aₙ): n∈ {0,...,m-1} and a₀,...,aₙ∈ [k]} where w is a word over k and (wₙ)ₙ₌₀ᵐ⁻¹ is a finite sequence of left variable words over k. We study the behavior of a family of measurable events in a probability space indexed by the elements of a Carlson-Simpson tree of sufficiently large dimension. Specifically we show the following. For every integer k≥ 2, every 0<ε≤ 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]<ⁿ of dimension at least Cor(k,ε,m) and every family {Aₜ:t∈ T} of measurable events in a probability space (Ω,Σ,μ) satisfying μ(Aₜ)≥ 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).