Ramsey theory over partitions I: Positive Ramsey relations from forcing axioms

TL;DR

This paper explores the relationship between anti-Ramsey properties and forcing axioms, establishing dualities and consistency results for positive Ramsey relations over partitions at uncountable cardinals.

Contribution

It uncovers a duality theorem linking anti-Ramsey properties and chain conditions, and shows positive Ramsey relations follow from forcing axioms at any uncountable cardinal.

Findings

Duality theorem under Martin's Axiom for uncountable cardinals

Consistency of positive Ramsey relations from forcing axioms without large cardinals

Resolution of two open problems from previous work (CKS21)

Abstract

In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over them is uncovered. At the level of the first uncountable cardinal this gives rise to a duality theorem under Martin's Axiom: a function $p:[\omega_1]^2\rightarrow\omega$ witnesses a weak negative Ramsey relation when $p$ plays the role of a coloring if and only if a positive Ramsey relation holds over $p$ when $p$ plays the role of a partition. The consistency of positive Ramsey relations over partitions does not stop at the first uncountable cardinal: it is established that at any prescribed uncountable cardinal these relations follow from forcing axioms without large cardinal strength. This result solves in particular two problems from [CKS21].