Rowbottom-type properties and a cardinal arithmetic

Abstract:

Assuming Rowbottom-type properties, we estimate the size of certain families of closed disjoint functions. We show that whenever $k$ is Rowbottom and ${2^\omega } < {\aleph _{{\omega _1}}}(k)$, then ${2^{ < k}} = {2^\omega }$ or $k$ is the strong limit cardinal. Next we notice that every strongly inaccessible Jónsson cardinal $k$ is $v$-Rowbottom for some $v < k$. In turn, Shelah’s method allows us to construct a Jónsson model of cardinality ${k^ + }$ provided ${k^{{\text {cf(}}k{\text {)}}}} = {k^ + }$. We include some additional remarks.