A partition property characterizing cardinals hyperinaccessible of finite type

Let P(n, (2) be the class of infinite cardinals which have the following property: Suppose for each v<KK that Cv is a partition of [KIn and card (C.)< K; then there is X C K of length a such that for each v < K, the set X (v + 1) is C,-homogeneous. In this paper the classes P (n, a) are studied and a nearly complete characterization of them is given. A principal result is that P (n + 2, n + 5) is the class of cardinals which are hyperinaccessible of type n. A partition property which differs from usual ones in that many partitions are considered simultaneously is defined and investigated in this paper. This property is interesting because it leads to an elementary characterization of the class of cardinals which are hyperinaccessible of a given finite type. The motivation for such a characterization comes from a problem in model theory. This problem is satisfactorily solved (as announced in [41 and [51) using some of the combinatorial results of this paper. These results lead naturally to a combinatorial problem which seems to be of sufficient independent interest so as to warrant further investigation. We give here an almost complete solution of this combinatorial problem; the final step for a complete solution seems elusive. This paper is a reworked version of Chapter 7 of my Ph. D. thesis [61 written under the supervision of Professor Robert L. Vaught. The other parts of my thesis, which consist of the model-theoretic applications of the results included here as well as generalizations of these results, will appear elsewhere. 1. The basic concepts. An ordinal number is the set of its predecessors. Ordinals are denoted by the Greek letters a, 3, y, v, e. Cardinal numbers are identified with initial ordinals and are denoted by K, A, li, where K is always an infinite cardinal. The symbols n, m, i always denote finite ordinals. A cardinal K is a strong limit cardinal iff 2\ < K whenever A < K. An inaccessible cardinal is a regular, strong limit cardinal. We need the concept of a Received by the editors June 17, 1971. AMS (MOS) subject classifications (1970). Primary 02K35, 04A10; Secondary 02H05, 02H13.