In this series of papers we advance Ramsey theory 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: [ω₁]² → ω 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 arbitrarily high uncountable cardinals these relations follow from forcing axioms without large cardinal strength. This result solves in particular two problems from [CKS21].

在本系列论文中，我们提出了关于分区的拉姆齐理论。在这一部分中，揭示了分区的反拉姆齐特性与使它们上的颜色均匀化的自然强迫概念的链条件之间的对应关系。在第一个不可数基数的层面上，这产生了马丁公理下的对偶定理：当p扮演着色角色时，函数p : [ ω ₁ ] ² → ω见证弱负 Ramsey 关系当且仅当当p充当划分角色时，拉姆齐关系在p上成立。

正拉姆齐关系在分区上的一致性并不仅限于第一个不可数基数：可以确定的是，在任意高的不可数基数上，这些关系是通过在没有大基数强度的情况下强制公理得出的。这个结果特别解决了[CKS21]中的两个问题。