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arXiv · 2204.14101

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

Menachem Kojman·Assaf Rinot·Juris Steprans
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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.

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Menachem Kojman, Assaf Rinot, Juris Steprans. 2022-04-29. Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems. https://arxiv.org/abs/2204.14101

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