A Polarized Partition Relation and Failure of Gch at Singular Strong Limit Sh586

The main result is that for λ strong limit singular failing the continuum hypothesis (i.e. 2 > λ), a polarized partition theorem holds. § 0. Introduction In the present paper we show a polarized partition theorem for strong limit singular cardinals λ failing the continuum hypothesis. Let us recall the following definition. {jeden} Definition 0.1. For ordinal numbers α1, α2, β1, β2 and a cardinal θ, the polarized partition symbol