PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 2, February 1999, Pages 577–581 S 0002-9939(99)04513-X OPEN COVERS AND PARTITION RELATIONS MARION SCHEEPERS (Communicated by Andreas R. Blass) Abstract. An open cover of a topological space is said to be an ω–cover if there is for each finite subset of the space a member of the cover which contains the finite set, but the space itself is not a member of the cover. We prove theorems which imply that a set X of real numbers has Rothberger’s property C 00 if, and only if, for each positive integer k, for each ω–cover U of X, and for each function f :[U ] 2 →{1,...,k} from the two-element subsets of U , there is a subset V of U such that f is constant on [V ] 2 , and each element of X belongs to infinitely many elements of V (Theorem 1). A similar characterization is given of Menger’s property for sets of real numbers (Theorem 6). Gerlits and Nagy [3] introduced the notion of an ω–cover as defined above in the abstract. Let Ω be the set of all ω–covers of X . These authors call X an –space if each ω–cover has a countable subset which is an ω–cover. All separable metric spaces are –spaces. An open cover U of X is said to be a large cover if for each x ∈ X the set {U ∈U : x ∈ U } is infinite. Let Λ be the set of all large covers and O the set of all open covers of X . For collections A and B of subsets of the set S the symbol S 1 (A, B) denotes the property that there is for each sequence (O n : n =1, 2, 3,... ) of elements of A a sequence (T n : n =1, 2, 3,... ) such that for each nT n ∈ O n , and {T n : n =1, 2, 3,... }∈B. Associated with this property we have the game G 1 (A, B): Two players, ONE and TWO, play an inning per positive integer. In the n–th inning ONE chooses a set O n ∈A, and TWO responds with T n ∈ O n . A play (O 1 ,T 1 ,O 2 ,T 2 ,... ) is won by TWO if {T n : n =1, 2, 3,... } is in B; otherwise, ONE wins. In the paper [9] where he introduced it, Rothberger used the symbol C 00 to denote the property S 1 (O, O). Galvin introduced the game G 1 (O, O) in [2] and Pawlikowski proved in [7]: Theorem 1 (Pawlikowski). X has property S 1 (O, O) if, and only if, ONE does not have a winning strategy in the game G 1 (O, O). For A and B as above the symbol S f in (A, B) denotes the property that there is for each sequence (O n : n =1, 2, 3,... ) of elements of A a sequence (T n : n = 1, 2, 3,... ) such that for each nT n is a finite subset of O n , and  ∞ n=1 T n ∈B. Received by the editors April 15, 1996 and, in revised form, May 16, 1997. 1991 Mathematics Subject Classification. Primary 03E05, 05D10. Key words and phrases. Ramsey’s theorem, Rothberger’s property, Menger’s property, infinite game, partition relation. The author’s research was funded in part by NSF grant DMS 95-05375. c 1999 American Mathematical Society 577 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use