Open covers and partition relations

An open cover of a topological space is said to be an $\omega$-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 $\textsf{C}^{\prime\prime}$ if, and only if, for each positive integer $k$, for each $\omega$-cover $\mathcal{U}$ of $X$, and for each function $f:[\mathcal{U}]^2\rightarrow\{1,\dots,k\}$ from the two-element subsets of $\mathcal{U}$, there is a subset $\mathcal{V}$ of $\mathcal{U}$ such that $f$ is constant on $[\mathcal{V}]^2$, and each element of $X$ belongs to infinitely many elements of $\mathcal{V}$ (Theorem 1). A similar characterization is given of Menger's property for sets of real numbers (Theorem 6).