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

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Abstract

In this series of papers we advance Ramsey theory 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 λ, a strong coloring on λ+ by λ colors over a partition can be stretched to one with λ+ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing Pr1(…) over a partition may be improved to witness Pr0(…).

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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Acknowledgments

Kojman was partially supported by the Israel Science Foundation (grant agreement 665/20). Rinot was partially supported by the Israel Science Foundation (grant agreement 2066/18) and by the European Research Council (grant agreement ERC-2018-StG 802756). Steprāns was partially supported by NSERC of Canada.

The authors are grateful to an anonymous referee, whose thoughtful comments and suggestions helped to improve the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er Sheva, 84105, Israel

    Menachem Kojman

  2. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel

    Assaf Rinot

  3. Department of Mathematics & Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

    Juris Steprāns

Authors
  1. Menachem Kojman
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  2. Assaf Rinot
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  3. Juris Steprāns
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Correspondence to Assaf Rinot.

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Kojman, M., Rinot, A. & Steprāns, J. Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2574-9

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  • DOI: https://doi.org/10.1007/s11856-023-2574-9

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