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 ℵ1. 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 Sierpi´ nski a hundred years ago.
| Original language | English |
|---|---|
| Pages (from-to) | 369-384 |
| Number of pages | 16 |
| Journal | Proceedings of the American Mathematical Society |
| Volume | 151 |
| Issue number | 1 |
| DOIs | |
| State | Published - 1 Jan 2023 |
Bibliographical note
Funding Information:Received by the editors March 31, 2021, and, in revised form, January 4, 2022, and March 30, 2022. 2020 Mathematics Subject Classification. Primary 03E02; Secondary 03E35, 03E17. The first author was partially supported by the Israel Science Foundation (grant agreement 665/20). The second author was partially supported by the Israel Science Foundation (grant agreement 2066/18) and by the European Research Council (grant agreement ERC-2018-StG 802756). The third author was partially supported by NSERC of Canada. 1Two d-tuples (p1,...,pd) and (q1,...,qd) are understood to be disjoint iff {p1,...,pd} ∩ {q1,...,qd}∕ ∅.
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© 2022 American Mathematical Society.