Ramsey theory over partitions III: Strongly Luzin sets and partition relations

Kojman, Menachem; Rinot, Assaf; Steprāns, Juris

doi:10.1090/proc/16106

Ramsey theory over partitions III: Strongly Luzin sets and partition relations
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by Menachem Kojman, Assaf Rinot and Juris Steprāns

Proc. Amer. Math. Soc. 151 (2023), 369-384

DOI: https://doi.org/10.1090/proc/16106

Published electronically: September 9, 2022

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Abstract:

The strongest type of coloring of pairs of countable ordinals, gotten by Todorčević from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size $\aleph _1$. In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it.

This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpiński a hundred years ago.

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