Suslin trees, the bounding number, and partition relations

Abstract

We investigate the unbalanced ordinary partition relations of the form $\lambda \rightarrow {(\lambda, \alpha)}^{2}$ for various values of the cardinal $\lambda$ and the ordinal $\alpha$. For example, we show that for every infinite cardinal $\kappa,$ the existence of a ${\kappa}^{+}-$Suslin tree implies ${\kappa}^{+} \not\rightarrow {\left( {\kappa}^{+}, {\log}_{\kappa}({\kappa}^{+}) + 2 \right)}^{2}$. The consistency of the positive partition relation $\mathfrak{b} \rightarrow {(\mathfrak{b}, \alpha)}^{2}$ for all $\alpha < {\omega}_{1}$ for the bounding number $\mathfrak{b}$ is also established from large cardinals.