Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems

TL;DR

This paper advances the understanding of anti-Ramsey relations over partitions by demonstrating how strong colorings can be extended and improved, generalizing classical pump-up theorems in Ramsey theory.

Contribution

It proves that strong colorings over partitions can be extended to larger color sets and provides conditions to upgrade certain colorings, generalizing existing pump-up theorems.

Findings

Strong colorings can be stretched to larger color sets over partitions.

Conditions are identified for upgrading from $Pr_1$ to $Pr_0$ colorings.

Results generalize classical pump-up theorems to partitioned settings.

Abstract

In this series of papers we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21]. It is shown that for every infinite cardinal $\lambda$, a strong coloring on $\lambda^+$ by $\lambda$ colors over a partition can be stretched to one with $\lambda^{+}$ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing $Pr_1(\ldots)$ over a partition may be improved to witness $Pr_0(\ldots)$. Since the classical theory corresponds to the special case of a partition with just one cell, the two results generalize pump-up theorems due to Eisworth and Shelah, respectively.