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1. Introduction. In this paper tie are going to give a classification of denumerable order types, namely we are going to prove that every order type of a denumerable set which does not contain a dense subset can be built up from the order types 0, 1 by a transfinite induction process taking at every step the so-called (o-sic-m and *-stem of order types previously defined. Thus every order type O of such kind can have an ordinal number o (O) less than o) 17 called the rank o f O, associated with it-and several properties of denumerable order types can be verified by carrying out a transfinite induction on Q(0) (1). As an application of the above-mentioned result a problem in the partition calculus for sets will be solved. Finally we are going to state some unsolved problems concerning non denumerable types (2). 2. Notations. Definitions. We are going to use the usual notations of set theory and we list only those where there is a danger of misunderstanding. denote non-negative integers .-No distinction will be made between finite cardinal numbers and ordinal numbers. 21 will denote the type of rational numbers ordered according to magn it ude. X, ~ denote the cardinal number of X and T' respectively. If S is a set ordered by a relation R, then for an arbitrary pair x, y E S "x is less than y" will be denoted by x < y (R) and the order type of X will be denoted by X (R). If there is no danger of misunderstanding (R) will be omitted. (1) This classification seems to be so simple and natural that probably it is already described somewhere in the literature ; however, the authors have been unable to find it. Therefore it seems worthwhile to rive the proofs in detail. (3) For another application of the classification see (1] .

On a classification of denumerable order types and an application to the partition calculus