Journal of the Mathematical Society of Japan

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


Full-text: Open access


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

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

First available in Project Euclid: 26 April 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

subtle cardinals partition relations Ramsey theory large cardinals


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.

Export citation


  • Y. Abe, Notes on subtlety and ineffability in $P\sb \kappa\lambda$, Arch. Math. Logic, 44 (2005), 619–631.
  • P. Barendse, The composition of large cardinal axioms, preprint, 2009.
  • J. E. Baumgartner, Ineffability properties of cardinals, I, In: Infinite and finite sets(Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), I, Colloq. Math. Soc. János Bolyai, 10, North-Holland, Amsterdam, 1975, pp.,109–130.
  • D. M. Carr, The minimal normal filter on $P\sb{\kappa }\lambda $, Proc. Amer. Math. Soc., 86 (1982), 316–320.
  • D. M. Carr, $P\sb \kappa\lambda$ partition relations, Fund. Math., 128 (1987), 181–195.
  • D. M. Carr, J.-P. Levinski and D. H. Pelletier, On the existence of strongly normal ideals over $P\sb \kappa\lambda$, Arch. Math. Logic, 30 (1990), 59–72.
  • P. Erdős and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc., 62 (1956), 427–489.
  • T. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic, 5 (1972/73), 165–198.
  • A. Kanamori, On Vop\v enka's and related principles, In: Logic Colloquium '77, Proc. Conf., Wrocław, 1977, Stud. Logic Foundations Math. 96, North-Holland, Amsterdam, 1978, pp.,145–153.
  • A. Kanamori, Regressive partition relations, $n$-subtle cardinals, and Borel diagonalization, In: International Symposium on Mathematical Logic and its Applications, Nagoya, 1988, Ann. Pure Appl. Logic, 52 (1991), 65–77.
  • M. Magidor, Combinatorial characterization of supercompact cardinals, Proc. Amer. Math. Soc., 42 (1974), 279–285.
  • T. K. Menas, On strong compactness and supercompactness, Ann. Math. Logic, 7 (1974/75), 327–359.
  • T. Usuba, private communication.