KNASTER AND FRIENDS III: SUBADDITIVE COLORINGS

Chris Lambie-Hanson, Assaf Rinot

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals θ < k, the existence of a strongly unbounded coloring c: [k2] → θ is a theorem of ZFC. Adding the requirement of subadditivity to a strongly unbounded coloring is a significant strengthening, though, and here we see that in many cases the existence of a subadditive strongly unbounded coloring c: [k2] → θ is independent of ZFC. We connect the existence of subadditive strongly unbounded colorings with a number of other infinitary combinatorial principles, including the narrow system property, the existence of k-Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring c: [k2] → θ is equivalent to a certain weak indexed square principle ind(k, θ). We conclude the paper with an application to the failure of the infinite productivity of -stationarily layered posets, answering a question of Cox.

Original languageEnglish
Pages (from-to)1230-1280
Number of pages51
JournalJournal of Symbolic Logic
Volume88
Issue number3
DOIs
StatePublished - 30 Sep 2023

Bibliographical note

Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic.

Funding

FundersFunder number
Horizon 2020 Framework Programme802756

    Keywords

    • Aronszajn tree
    • ascent path
    • coherent coloring
    • indexed square
    • stationarily layered posets
    • strongly unbounded coloring
    • subadditive coloring

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