Measurable events indexed by words

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).