Authors: Di Prisco, Carlos Augusto
Todorčević, Stevo 
Title: Souslin partitions of products of finite sets
Journal: Advances in Mathematics
Volume: 176
Issue: 1
First page: 145
Last page: 173
Issue Date: 1-Jun-2003
Rank: M21a
ISSN: 0001-8708
DOI: 10.1016/S0001-8708(02)00064-6
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).
Publisher: Elsevier
Project: CNRS (France)—CONICIT (Venezuela) cooperation agreement, Project CNRS 10062—CONICIT PI 2000001471

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