A partition property characterizing cardinals hyperinaccessible of finite type

Abstract

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