Borel partitions of products of finite sets

Abstract

In this paper we prove the following polarized partition relation: For every sequence {mt}i<ω of natural numbers, there is an increasing sequence {ni}t<ω such that: for every sequence of sets {Ai}t<ω with |At| = nt, and for every Borel partition f : (Πt<ω At) → 2, there are sets {Ht}t<ω with |Ht| = mt such that f is constant on (Πt<ω Ht). This gives a positive answer to the Borel version of the following question asked in [DiPH]. Is there a sequence {ni}t<ω of natural numbers such that the partition relation (Formula Presented) holds?