A polarized partition relation and failure of GCH

A proof of Shelah's recent partition theorem

Shelah's own proof to his recent polarized partition theorem involving a singular strong limit that violates the GCH is presented. The proof is slightly re-arranges so that no use of the ideal I[λ] is made. The proof should be readabel to any student with knowledge of set theory.

Open and solved problems concerning polarized partition relations

We list some open problems, concerning the polarized partition relation. We solve a couple of them by showing that for every singular cardinal $μ$ one can force the strong polarized relation with respect to the pair $μ^+,μ$.

关于极化划分关系的等价性

On equivalences of polarized partition relations

本文引入广义强序列乘积概念，并纯组合地证明其存在性与极化划分关系等价。

关于极化分拆关系的若干注记

Some remarks on polarized partition relations

本文引入广义强序列乘积的概念，通过纯组合证明其存在性可推导出极化分拆关系，拓展了Efimov提出的强序列工具的应用场景。

Consistency of partition relation for cardinal in (lambda,2^lambda)

We present a forcing for blowing up 2^lambda and making ``many positive polarized partition relations'' (in a sense made precise in (c) of our main theorem) hold in the interval [lambda, 2^lambda]. This generalizes results of [276], Section 1, and the forcing is a ``many cardinals'' version of the forcing there.

Partition Calculus and Cardinal Invariants

We prove that the strong polarized relation of $θ$ above $ω$ applied simultaneously for every cardinal in the interval $[\aleph_1,\aleph]$ is consistent. We conclude that this positive relation is consistent for every cardinal invariant on the continuum. We show that similar results hold for a supercompact cardinal, and for a strong limit singular under some assumptions.