Souslin partitions of products of finite sets

Abstract

To every infinite sequence of positive integers m = {mi : i∈ω}, we associate two fields of sets, a field ℂL(m) of subsets of ωω and a field ℙℂL(m) of subsets of ωω x [ω]ω. Their relevance to Ramsey theory is based on the fact that for every ℂL(m)-measurable partition c : ωω → 2 there is a sequence {Hi: i∈ω} with Hi = mi such that c is constant on ∏iεω Hi; similarly, for every ℙℂL(m)-measurable partition c : ωω x [ω]ω → 2 there is H ∈ [ω]ω and a sequence {Hi: i∈ω} of sets with Hi⊆ω and Hi = mi such that c is constant on (∏i∈ω Hi) x [H]ω. In Di Prisco et al. (J. Combin. Theory Ser. A 93 (2001) 333; Combinatorica, to appear) it is shown that ℂL(m) and ℙℂL(m) are σ-fields that contain all closed sets, and therefore all Borel subsets of their corresponding domains. We show here that they are in fact closed under Souslin operation, and that under suitable assumptions, they contain all reasonably definable subsets of their corresponding domains. These results are then used to show that the classical partition relation ω→ (ω )ω is not equivalent to its polarized version, solving thus a long-standing problem in this area (see J. Symbolic Logic 58 (1998) 860; Notas de Lógica Matematica, Vol. 39, Universidad Nacional del Sur, Bahía Blanca, Argentina, 1994, pp. 89-94).