MATHEMATICS DISCRETE SUBSPACES OF TOPOLOGICAL SPACES 1 ) BY A. HAJNAL AND I. JUHASZ (Communicated by Prof. J. POPKEN at the meeting of October 29, 1966) §. l. Introduction Recently several papers appeared in the literature proving theorems of the following type. A topological space with "very many points" contains a discrete subspace with "many points". J. DE GROOT and B. A. EFIMOV proved in [2] and [4] that a Hausdorff space of power > exp exp exp m contains a discrete subspace of power >m. J. ISBELL proved in [3] a similar result for completely regular spaces. J. de Groot proved as well that for regular spaces R the assumption IRI > exp exp m is sufficient to imply the existence of a discrete subspace of potency > m. One of our main issues will be to improve this result and show that the same holds for Hausdorff spaces (see Theorems 2 and 3). Our Theorem 1 states that a Hausdorff space of density > exp m contains a discrete subspace of power >m. We give two different proofs for the main result already mentioned. The proof outlined for Theorem 3 is a slight improvement of de Groot's proof. The proof given for Theorem 2 is of purely combinatorial character. We make use of the ideas and some theorems of the so called set-theoretical partition calculus developed by P. ERDOS and R. RADO (see [5], [ 11 ]). Almost all the other results we prove are based on combinatorial theorems. For the convenience of the reader we always state these theorems in full detail. Our Theorem 4 states that if m is a strong limit cardinal which is the sum of No smaller cardinals, then every Hausdorff space of power m contains a discrete subspace of power m. The problem if the same holds for all strong limit cardinals remains open. At the end of § 4 using the generalized continuum hypothesis (G.C.H. in what follows) we give a discussion of the results and problems. In § 5 we consider the problem of existence of large discrete subspaces under additional assumptions. Theorem 5 states that a Hausdorff space of power > 2m contains > m disjoint open sets provided the character of the space is at most m. I) A preliminary report containing the main results of this paper appeared in the Doklady Akad. Nauk. SSSR (see [1]).