Matrices of completely Ramsey sets with infinitely many rows



The main result of the present article is the following: Let N be an infinite subset of math formula, math formula, math formula and let math formula math formula math formula be a matrix with infinitely many rows of completely Ramsey subsets of math formula such that for every n, math formula. Then there exist math formula, a sequence math formula of nonempty finite subsets of N, and an infinite subset T of math formula such that math formula for every infinite subset I of math formula. We also give an application of this result to partitions of an uncountable analytic subset of a Polish space X into sets belonging to the σ-algebra generated by the analytic subsets of X.