## Journal of the Mathematical Society of Japan

### Necessary and sufficient conditions for the existence of an n-subtle cardinal

Peter BARENDSE

#### Abstract

We extend the work of Abe in [1], to show that the strong partition relation $C \rightarrow (n+2)^{n+1}_{<-reg}$, for every $C \in \mathsf{WNS}^{*}_{\kappa,\lambda}$, is a consequence of the existence of an n-subtle cardinal. We then build on Kanamori's result in [10], that the existence of an $n$-subtle cardinal is equivalent to the existence of a set of ordinals containing a homogeneous subset of size $n$+2 for each regressive coloring of $n$+1-tuples from the set. We use this result to show that a seemingly weaker relation, in the context of $P_{\kappa}\lambda$ is also equivalent. This relation is a new type of regressive partition relation, which we then attempt to characterize.

#### Article information

Source
J. Math. Soc. Japan, Volume 64, Number 2 (2012), 489-506.

Dates
First available in Project Euclid: 26 April 2012

https://projecteuclid.org/euclid.jmsj/1335444400

Digital Object Identifier
doi:10.2969/jmsj/06420489

Mathematical Reviews number (MathSciNet)
MR2916076

Zentralblatt MATH identifier
1250.03109

Subjects
Primary: 03E55: Large cardinals
Secondary: 03E02: Partition relations

#### Citation

BARENDSE, Peter. Necessary and sufficient conditions for the existence of an n -subtle cardinal. J. Math. Soc. Japan 64 (2012), no. 2, 489--506. doi:10.2969/jmsj/06420489. https://projecteuclid.org/euclid.jmsj/1335444400

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