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Souslin partitions of products of finite sets

2003; Elsevier BV; Volume: 176; Issue: 1 Linguagem: Inglês

10.1016/s0001-8708(02)00064-6

1090-2082

Carlos Augusto Di Prisco, Stevo Todorčević,

Limits and Structures in Graph Theory

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×[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 ∏i∈ωHi; similarly, for every PCL(m→)-measurable partition c:ωω×[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∈ωHi)×[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, Bahı́a Blanca, Argentina, 1994, pp. 89–94).