Measurable events indexed by words

For every integer k ⩾ 2 let [ k ] < N be the set of all words over k. A Carlson–Simpson tree of [ k ] < N of dimension m ⩾ 1 is a subset of [ k ] < N of the form { w } ∪ { w ⌢ w 0 ( a 0 ) ⌢ … ⌢ w n ( a n ) : n ∈ { 0 , … , m − 1 } and a 0 , … , a n ∈ [ k ] } where w is a word over k and ( w n ) n = 0 m − 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. ery 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 { A t : t ∈ T } of measurable events in a probability space ( Ω , Σ , μ ) satisfying μ ( A t ) ⩾ 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 A t ) ⩾ θ ( k , ε , | F | ) . oof 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 ) .