Borel partitions of products of finite sets and the Ackermann function

Abstract

It is shown that for every primitive recursive sequence mi∞i=0 of positive integers, there is an ackermannic sequence ni∞i=0 of positive integers such that for every partition of the product ∏∞i=0ni into two Borel pieces, there are sets Hi⊆ni with Hi=mi such that the subproduct ∏∞i=0Hi is included in one of the pieces.