Joint work with Menachem Kojman and Juris Steprāns.
Abstract. The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic 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 Sierpinski a hundred years ago.
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Submitted to Proceedings of the American Mathematical Society, April 2021.
Accepted, April 2022.
In footnote #1, the definition of “disjoint” should say that the intersection IS empty.