The least cardinal for which the Baire category theorem fails is equal to the least cardinal for which a Ramseyan theorem fails. The Baire category theorem states that the real line is not the union of countably many meager (also known as first category) sets. Let cov(M) denote the least cardinal number such that there are that many first category subsets of the real line whose union is the entire real line. Then cov(M) is the least cardinal number for which the Baire category theorem fails. This cardinal number, defined in terms of topological notions, appears in many different guises in combinatorial set theory. A long list of diverse guises of this cardinal number is already general knowledge for set theorists; Galvin gave a game-theoretic version (part of which is published in [4], and part of which is unpublished—however, see [8]), A. W. Miller gave a characterization in terms of sequences of positive integers [6], which was later given an elegant improvement by Bartoszynski [1]. It is also known to be the least cardinal number such that there is a set of real numbers of that cardinality which does not have Rothberger’s property C′′. A set X of real numbers has property C′′ if, for every sequence (Un : n = 1, 2, 3, . . . ) of open covers of X , there is a sequence (Un : n = 1, 2, 3, . . . ) such that, for each n, Un ∈ Un and {Un : n = 1, 2, 3, . . .} is a cover for X . It seems that for the purposes of applications of set theory to other areas of mathematics, it would be useful to have as many non-trivial characterizations of this cardinal number as possible. In this paper we give a few more equivalent forms of this cardinal number. To explain some of our results, we need some terminology which is well-known in other contexts. Let κ be an infinite cardinal number which will be fixed for the duration of the paper. A collection of subsets of κ is said to be a cover of κ if its union is equal to κ. We shall be interested in countable covers of κ. A cover of κ is said to be an ω-cover if it is countably infinite, κ itself is not a member of the cover, and if there is for every finite subset of κ an element of this cover which contains it. We shall let the symbol Ω denote the collection of ω-covers of κ. Borrowing from Ramsey theory (see Section 8 of [3]), we shall use the symbol