Regressive partition relations for infinite cardinals

The regressive partition relation, which turns out to be important in incompleteness phenomena, is completely characterized in the transfinite case. This work is related to Schmerl [SI, whose characterizations we complete. Regressive functions arise naturally in the study of infinite cardinals, from Fodor's well-known lemma to contexts involving large cardinals (for example, the n-subtle cardinals of Baumgartner [B2]). In Kanamori and McAloon [KM], a regressive function version of the theorem of Erd6s and Rado [ER] on canonical partitions was miniaturized and shown to be independent of Peano arithmetic. This result in turn reverberated to the infinite context to raise new questions; here we completely characterize the corresponding partition symbol for infinite cardinals. In contrast to the Erdos-Rado Theorem for the ordinary partition symbol, we show that these partition relations actually provide a characterization of cardinals in the finite Mahlo hierarchy. Thus, just as with the finite miniaturization, an elementary combinatorial property leads to a necessary transcendence. Our work confirms some speculations in McAloon [M], where an infinitary analogue of the Paris-Harrington partition relation is considered. After we had already established some characterizations, we became aware of the close relationship of this work to the results of Schmerl [S]. The third author then saw how to sharpen the characterization of Schmerl's property as well as ours, and this paper is written so as to approach these optimal results most directly. The sharpening uses ideas of Todorcevic [T] who noted that our 3.4 for n = 0 can be derived directly from his work. In ?1, we begin the study of our partition symbol and establish the straightforward positive results about Mahlo cardinals. In ?2, we develop some technical formulations and lemmata. Finally, we apply this work in ?3 to establish the optimal results on the necessity of Mahlo cardinals. We discuss the connections with Schmerl [S] at the end. We would like to thank Jim Schmerl for several expository suggestions, particularly for providing the statement of 3.3. 1. Preliminaries. We first formulate the regressive partition symbol: Let X be a set of ordinals and n a natural number. If f is a function with domain [XI n , we write f(ao,.. ., an-1) for f({ao, ... , an-1}), with the understanding that ao < *** K ?n-1* Such a function is called regressive if f(a0o,..., an-i) < ao Received by the editors December 26, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 03E05; Secondary 03E55.