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

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].