Stationary and closed rainbow subsets

Garti, Shimon; Zhang, Jing

doi:10.1016/j.apal.2020.102887

Annals of Pure and Applied Logic

2021

DOI: 10.1016/j.apal.2020.102887

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Stationary and closed rainbow subsets

Shimon Garti

Jing Zhang

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2022

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Some remarks on uncountable rainbow Ramsey theory

Zhang

2022

Proc. Amer. Math. Soc.

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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 \kappa , κ → p o l y ( κ ) 2 − b d d 2 \kappa \to ^{poly}(\kappa )^2_{2-bdd} does not characterize weak compactness and for any singular cardinal κ \kappa , ◻ κ \square _\kappa implies κ + ↛ p o l y ( η ) > κ − b d d 2 \kappa ^+\not \to ^{poly} (\eta )^2_{>\kappa -bdd} for any η ≥ c f ( κ ) + \eta \geq cf(\kappa )^+ and κ > c f ( κ ) = κ \kappa ^{>cf(\kappa )}=\kappa implies κ + → p o l y ( ν ) > κ − b d d 2 \kappa ^+\to ^{poly} (\nu )^2_{>\kappa -bdd} for any ν > c f ( κ ) + \nu >cf(\kappa )^+ . We also provide a simplified construction of a model for ω 2 ↛ p o l y ( ω 1 ) 2 − b d d 2 \omega _2\not \to ^{poly} (\omega _1)^2_{2-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.

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Some remarks on uncountable rainbow Ramsey theory

Zhang

2022

Proc. Amer. Math. Soc.

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Stationary and closed rainbow subsets

Cited by 1 publication

References 12 publications

Some remarks on uncountable rainbow Ramsey theory

Some remarks on uncountable rainbow Ramsey theory

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