Suslin trees, the bounding number, and partition relations

TL;DR

This paper explores complex partition relations in set theory, linking the existence of Suslin trees and large cardinals to specific unbalanced partition properties involving infinite cardinals and the bounding number.

Contribution

It establishes new connections between Suslin trees, the bounding number, and unbalanced partition relations, including consistency results under large cardinal assumptions.

Findings

Existence of a -Suslin tree implies certain negative partition relations.

Consistency of positive partition relations for the bounding number  established from large cardinals.

Results extend understanding of partition relations in the context of set-theoretic combinatorics.

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.