Open covers and partition relations

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″ 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 / 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).