Open and solved problems concerning polarized partition relations

Abstract

Abstract.We list some open problems, concerning the polarized partition relation. We solve a couple of them, by showing that for every limit non-inaccessible ordinal α there exists a forcing notion P such that the strong polarized relationholds in V P . Nous passons en revue certains problèmes non résolus conceront la relation de partition polarisée. Nous en résolvons deux en montrant que pour chaque ordinal limite non inaccessible α, il existe un forcing P tel que V P satisfait la relation polarisée forte2010 Mathematics Subject Classification. 03E05, 03E35.

Random reals and polarized colorings

“…2 for every µ ∈ (ℵ 0 , λ], as shown in [6], Remark 2.4. In particular, it holds for µ = c. As r = c in this generic extension and λ > ℵ 1 we have the consistency of the negative direction.…”

“…If λ ≥ κ are infinite cardinals then the strong polarized relation λ κ → λ κ 1,1 2 means that for every c : λ×κ → 2 there are A ∈ [λ] λ , B ∈ [κ] κ such that c ↾ (A×B) is constant. We shall make use of the following theorem from [6]: 0.4 Observe that the requirement about the cofinality of κ is stronger in this theorem, and we do not have full knowledge when ℵ 0 < cf(κ) ≤ r, see below.…”

“…If λ ≥ κ are infinite cardinals then the strong polarized relation λ κ → λ κ 1,1 2 means that for every c : λ×κ → 2 there are A ∈ [λ] λ , B ∈ [κ] κ such that c ↾ (A×B) is constant. We shall make use of the following theorem from [6]…”

Cardinal characteristics of the continuum and partitions

“…2 (and in fact something a bit stronger, increasing the number of colours and obtaining a negative square-bracket relation). As corollaries, we answer several questions from [016GS].…”

“…Theorem 2.2 solves[016GS, Problem 3.19]. The problem asks whether for a cardinal ℵ 1 < κ < c it is possible for κ ω to be destroyed by the Lévy collapse of c to κ.…”