## Journal of the Mathematical Society of Japan

### Two-cardinal versions of weak compactness: Partitions of triples

#### Abstract

Let κ be a regular uncountable cardinal, and λ be a cardinal greater than κ. Our main result asserts that if (λ< κ)<(λ< κ) = λ< κ, then (pκ, λ(NInκ, λ< κ))+ $\longrightarrow$ ((NSκ, λ[λ]< κ)+, NSκ, λs+)3 and (pκ, λ(NAInκ, λ< κ))+ $\longrightarrow$ (NSκ, λs+)3, where NSκ, λs (respectively, NSκ, λ[λ]< κ) denotes the smallest seminormal (respectively, strongly normal) ideal on Pκ (λ), NInκ, λ< κ (respectively, NAInκ, λ< κ) denotes the ideal of non-ineffable (respectively, non-almost ineffable) subsets of Pκ< κ), and pκ, λ: Pκ< κ) → Pκ (λ) is defined by pκ, λ(x) = x ∩ λ.

#### Article information

Source
J. Math. Soc. Japan, Volume 67, Number 1 (2015), 207-230.

Dates
First available in Project Euclid: 22 January 2015

https://projecteuclid.org/euclid.jmsj/1421936551

Digital Object Identifier
doi:10.2969/jmsj/06710207

Mathematical Reviews number (MathSciNet)
MR3304020

Zentralblatt MATH identifier
1330.03080

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

#### Citation

MATET, Pierre; USUBA, Toshimichi. Two-cardinal versions of weak compactness: Partitions of triples. J. Math. Soc. Japan 67 (2015), no. 1, 207--230. doi:10.2969/jmsj/06710207. https://projecteuclid.org/euclid.jmsj/1421936551

#### References

• Y. Abe, Saturation of fundamental ideals on $\mathscr{P}_\kappa \lambda$, J. Math. Soc. Japan, 48 (1996), 511–524.
• Y. Abe, A hierarchy of filters smaller than ${\rm CF}_{\kappa\lambda}$, Arch. Math. Logic, 36 (1997), 385–397.
• Y. Abe, Combinatorial characterization of $\Pi^1_1$-indescribability in $P_\kappa\lambda$, Arch. Math. Logic, 37 (1998), 261–272.
• Y. Abe, Notes on subtlety and ineffability in $P_\kappa \lambda$, Arch. Math. Logic, 44 (2005), 619–631.
• Y. Abe and T. Usuba, Notes on the partition property of $\mathcal{P}_\kappa \lambda$, Arch. Math. Logic, 51 (2012), 575–589.
• J. E. Baumgartner, Ineffability properties of cardinals. I, In: Infinite and Finite Sets, vol.,I, Keszthely, 1973, (eds. A. Hajnal, R. Rado and Vera T. Sós), Colloq. Math. Soc. Janos Bolyai, 10, North-Holland, Amsterdam, 1975, pp.,109–130.
• D. M. Carr, The structure of ineffability properties of $P_\chi \lambda$, Acta Math. Hungar., 47 (1986), 325–332.
• D. M. Carr, $P_\kappa\lambda$ 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 $P_\kappa\lambda$, Arch. Math. Logic, 30 (1990), 59–72.
• T. J. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic, 5 (1973), 165–198.
• C. A. Johnson, More on distributive ideals, Fund. Math., 128 (1987), 113–130.
• C. A. Johnson, Some partition relations for ideals on $P_\chi \lambda$, Acta Math. Hungar., 56 (1990), 269–282.
• S. Kamo, Ineffability and partition property on $\mathscr{P}_\kappa\lambda$, J. Math. Soc. Japan, 49 (1997), 125–143.
• M. Magidor, Combinatorial characterization of supercompact cardinals, Proc. Amer. Math. Soc., 42 (1974), 279–285.
• P. Matet, Un principe combinatoire en relation avec l'ultranormalité des idéaux, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 61–62.
• P. Matet, ${\rm Part}(\kappa,\lambda)$ and ${\rm Part}^\ast(\kappa,\lambda)$, In: Set Theory, Barcelona, 2003–2004, (eds. J. Bagaria and S. Todorcevic), Trends Math., Birkhäuser, Basel, 2006, 321–344.
• P. Matet, Covering for category and combinatorics on $P_\kappa(\lambda)$, J. Math. Soc. Japan, 58 (2006), 153–181.
• P. Matet, Large cardinals and covering numbers, Fund. Math., 205 (2009), 45–75.
• P. Matet, Normal restrictions of the noncofinal ideal on $P_\kappa(\lambda)$, Fund. Math., 221 (2013), 1–22.
• P. Matet, C. Péan and S. Shelah, Confinality of normal ideals on $P_\kappa(\lambda)$. II, Israel J. Math., 150 (2005), 253–283.
• P. Matet, C. Péan and S. Shelah, Confinality of normal ideals on $P_\kappa(\lambda)$. I, preprint..
• P. Matet and S. Shelah, The nonstationary ideal on $P_\kappa(\lambda)$ for $\lambda$ singular, preprint..
• P. Matet and T. Usuba, Two-cardinal versions of weak compactness: Partitions of pairs, Ann. Pure Appl. Logic, 163 (2012), 1–22.
• I. Neeman, Aronszajn trees and failure of the singular cardinal hypothesis, J. Math. Log., 9 (2009), 139–157.
• S. Shelah, Weakly compact cardinals: a combinatorial proof, J. Symbolic Logic, 44 (1979), 559–562.
• E. Specker, Sur un problème de Sikorski, Colloquium Math., 2 (1949), 9–12.
• T. Usuba, Ineffability of $\mathscr{P}_\kappa\lambda$ for $\lambda$ with small cofinality, J. Math. Soc. Japan, 60 (2008), 935–954.