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# Rowbottom-Type Properties and a Cardinal Arithmetic

Jan Tryba
Proceedings of the American Mathematical Society
Vol. 96, No. 4 (Apr., 1986), pp. 661-667
DOI: 10.2307/2046322
Stable URL: http://www.jstor.org/stable/2046322
Page Count: 7
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## Abstract

Assuming Rowbottom-type properties, we estimate the size of certain families of closed disjoint functions. We show that whenever κ is Rowbottom and $2^\omega < \aleph_{\omega_1}(\kappa)$, then $2^{<\kappa} = 2^\omega$ or κ is the strong limit cardinal. Next we notice that every strongly inaccessible Jónsson cardinal κ is ν-Rowbottom for some $\nu < \kappa$. In turn, Shelah's method allows us to construct a Jónsson model of cardinality κ+ provided $\kappa^{\operatorname{cf}(\kappa)} = \kappa^+$. We include some additional remarks.

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