Suslin trees, the bounding number, and partition relations

Abstract

We investigate the unbalanced ordinary partition relations of the form $\lambda \to {(\lambda ,\alpha )}^2$ 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 ${\kappa }^{+}\nrightarrow {\left({\kappa }^{+},{\log }_{\kappa }({\kappa }^{+})+2\right)}^2$. The consistency of the positive partition relation $\mathfrak{b}\to {(\mathfrak{b},\alpha )}^2$ for all $\alpha <{\omega }_1$ for the bounding number $\mathfrak{b}$ is also established from large cardinals.

arXiv comment17 pages, submitted