Abstract

We present two different types of models where, for certain singular cardinals λ of uncountable cofinality, λ → (λ,ω + 1) 2 , although λ is not a strong limit cardinal. We announce, here, and will present in a subsequent paper, [7], that, for example, consistently, $\aleph_{\omega_1} \nrightarrow (\aleph_{\omega_1}, \omega + 1)^2$ and consistently, 2 $^{\aleph_0} \nrightarrow (2^{\aleph_0},\omega + 1)^2$