Weak Partition Properties for Infinite Cardinals. I

Partition properties are perhaps the most fruitful of the various methods for defining and discussing large cardinals in set theory. In this paper we weaken in a natural way the most well known of these partition properties and examine the extent to which the cardinals defined remain "large." 1. 1.1. An area of set theory which has come under a great deal of study recently is that concerned with partition properties for cardinal numbers. In order to consider several specific examples let k denote an uncountable cardinal and let us introduce k—*(k)2 (k—*(k)(k)(k)<¡°, then there exists a nonconstructible A3 set of integers. R3 If k—>(k)2, then k is a strongly inaccessible cardinal. R4 For X an infinite cardinal less than k, let k—>(k)1 (k—*ík)\") denote the assertion (*) changed to cover partitions into X pieces. Then for each infinite cardinal X less than k, k—>(k)x («-»Wx"") is equivalent to k—*(k)2 (k—>(k)(k)2 and k—»(/t)(k)1 states that there exists a set x of cardinality k all of whose 2 element subsets are contained in only one of the pieces— what if we just assert that there exists a subset x of k of cardinality k all of whose 2 element subsets are contained in fewer than X of the pieces? We make this notion precise by introducing k^>(k)2 (kAOO^) to denote for each infinite cardinal X less than k and partition of the 2 Received by the editors March 13, 1968 and, in revised form, January 24, 1970. AMS 1970 subject classifications. Primary 02K35.