Ramsey theory over partitions I: Positive Ramsey relations from forcing axioms

Menachem Kojman, Assaf Rinot, Juris Steprāns

Research output: Contribution to journalArticlepeer-review

Abstract

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: [ω 1]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].

Original languageEnglish
JournalIsrael Journal of Mathematics
DOIs
StateAccepted/In press - 2023

Bibliographical note

Publisher Copyright:
© 2023, The Hebrew University of Jerusalem.

Funding

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.

FundersFunder number
Natural Sciences and Engineering Research Council of Canada
European Research CouncilERC-2018-StG 802756
Israel Science Foundation2066/18, 665/20

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