Parametrized Partitions of Products of Finite Sets*

For every infinite sequence of positive integers ${\left\{ {m_{i} } \right\}}^{\infty }_{{i = 0}}$ and every Borel partition c : ω ^ω×[ω]^ω→{0, 1} there is H∈[ω]^ω and a sequence ${\left\{ {H_{i} } \right\}}^{\infty }_{{i = 0}}$ of subsets of ω, with |H _i |=m _i for every i, such that c is constant on ${\left( {{\prod\nolimits_{i = 0}^\infty {H_{i} } }} \right)}x{\left[ H \right]}^{\omega }$.