Ordinary Partition Relations for Ordinal Numbers

To RICHARD RADO for his 65th birthday Q 1. Introduction The ordinary partition symbol invented by R. RADO and first introduced in [l] enables us to study systematically the possible generalizations of RAIXSEY'S theorem. denotes that the following statement is true. Let (S,-<) be an ordered set; tp S(-<) = ,z or let S be a set 1 S 1 = a if CL is an order type or z is a cardinal respectively. Let [Sir = {X : X c S// A I X I = r} = u (v < Y) J, b e an arbitrary r-partition of length y of S. Then there exist a subset S' c S and an ordinal v < y such that of the above statement is true. In [ 1 ], [2] and [3] several generalisations of (1. I) had been defined and a general partition calculus had been developed. In [3] an almost complete discussion of (1.1) is given in case the entries a, PO , ,. . , &, are cardinals and G. C. H. is assumed. In a forthcoming book of R. Rado and the authors this discussion wiI1 be given without using G. C. H. In this paper we will consider some special problems for the ordinary partition relation in case the entries are ordinals. We will only consider the case r = 2, and in most of the cases we assume y = 2 too. Even the problems concerning these special cases are rather ramified. In our paper [4] we gave a collect,ion of t'ypical unsolved problems. Here we will consider only one type of these problems.