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

Joint work with Menachem Kojman and Juris Steprāns.

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 [CKS20].

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.

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