It is shown that for every primitive recursive sequence {m(i)} (infinity)(i)(=0) of positive integers. there is an ackermannic sequence {n(i)} (infinity)(i)(=0) positive integers such that for every partition of the product Pi (infinity)(i=0) into two Borel pieces, there are sets H-i subset of or equal to n(i) with H-i = m(i) such that the subproduct Pi (infinity)(i=0) H-i is included in one of the pieces. (C) 2001 Academic Press. [References: 11]
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