Abstract
We discuss the rainbow Ramsey theorems at limit cardinals and successors of singular cardinals, addressing some questions by Uri Abraham, James Cummings, and Clifford Smyth [J. Symbolic Logic 72 (2007), pp. 865–895] and Uri Abraham and James Cummings [Cent. Eur. J. Math. 10 (2012), pp. 1004–1016]. In particular, we show for inaccessible κ, κ →poly (κ)22−bdd does not characterize weak compactness and for any singular cardinal κ, □κ implies κ+ ⇸poly (η)2<κ−bdd for any η ≥ cf(κ)+ and κ<cf(κ) = κ implies κ+ →poly (ν)2<κ−bdd for any ν < cf(κ)+. We also provide a simplified construction of a model for ω2 ⇸poly (ω1)22−bdd originally constructed by Uri Abraham and James Cummings [Cent. Eur. J. Math. 10 (2012), pp. 1004–1016] and show the witnessing coloring is indestructible under strongly proper forcings but destructible under some c.c.c forcing. Finally, we conclude with some remarks and questions on possible generalizations to rainbow partition relations for triples.
| Original language | English |
|---|---|
| Pages (from-to) | 865-880 |
| Number of pages | 16 |
| Journal | Proceedings of the American Mathematical Society |
| Volume | 151 |
| Issue number | 2 |
| DOIs | |
| State | Published - 1 Feb 2023 |
Bibliographical note
Publisher Copyright:© 2022 American Mathematical Society.
Funding
The work was done when I was a graduate student at Carnegie Mellon University supported in part by the US tax payers. Part of the revision was done during my time as a postdoctoral fellow at Bar-Ilan University, supported by the Foreign Postdoctoral Fellowship Program of the Israel Academy of Sciences and Humanities and by the Israel Science Foundation (grant agreement 2066/18).
| Funders | Funder number |
|---|---|
| Carnegie Mellon University | |
| Israel Academy of Sciences and Humanities | |
| Israel Science Foundation | 2066/18 |