Borel Partitions of Products of Finite Sets and the Ackermann Function

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=0 ni into two Borel pieces, there are sets Hi⊆ni with |Hi|=mi such that the subproduct ∏∞i=0 Hi is included in one of the pieces.