A partition property characterizing cardinals hyperinaccessible of finite type

Abstract:

Let ${\mathbf {P}}(n,\alpha )$ be the class of infinite cardinals which have the following property: Suppose for each $\nu < \kappa$ that ${C_\nu }$ is a partition of ${[\kappa ]^n}$ and card $({C_\nu }) < \kappa$; then there is $X \subset \kappa$ of length $\alpha$ such that for each $\nu < \kappa$, the set $X - (\nu + 1)$ is ${C_\nu }$-homogeneous. In this paper the classes ${\mathbf {P}}(n,\alpha )$ are studied and a nearly complete characterization of them is given. A principal result is that ${\mathbf {P}}(n + 2,n + 5)$ is the class of cardinals which are hyperinaccessible of type n.