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 , , and let be a matrix with infinitely many rows of completely Ramsey subsets of such that for every n, . Then there exist , a sequence of nonempty finite subsets of N, and an infinite subset T of such that for every infinite subset I of . 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.