In this paper we examine the relation between i?-sets, which are a purely set-theoretic concept, and various concepts associated with planar maps, for instance, four-colorings, five-colorings, Ήamiltonian circuits, and Petersen's theorem. Moreover, the introduction of the notion of a i?-set into graph theory enables us to ask questions which may be more tractable than the four-color conjecture and shed light on it. 1* Introduction and definitions* Let F be a family of sets. A set that meets every member of F and yet contains none of the members of F is called a i?-set for F. Observe that if B is a Bset for F, then so is its complement, (\JEeFE) — B. In fact, F has a £-set if and only if \JEeF E can be partitioned into two sets, A and B, such that neither A nor B contains a member of F. Observe that if F has a J5-set, and if G S F, then G has a 5-set. Also, if G C F, and every member of F contains some member of G, and if G has a i?-set, then F also has a B-set. (The notion of j?-set goes back to Bernstein, who used it in 1908 to deal with a topological question.) We shall be concerned with maps covering the surface of the sphere, S2. For the most part, we will assume that these maps are 3-regular, that is, each vertex has degree three. Each region of the map will be a topological cell. Two regions are adjacent if they share at least one edge. A sequence of distinct regions Ru R2, , Rn+1, n^>l such that Ri is adjacent to Ri+1, 1 ^ i ^ n is a path of regions. If we have Rn+1 = Rι, and Ru R2, •••!?» are still distinct, we call the path a region-cycle of length n (or n-r eg ion-cycle). A region-cycle consisting simply of the regions around a vertex we call a basic cycle. In a 3-regular map the basic cycles have length three. If the union of any two regions in a map is simply connected, then the regions bordering any given region form a region-cycle, which we call a face cycle. Its length is just the number of edges of the surrounded region. We shall not be interested in region-cycles of length two, unless their union is not simply connected. A region-cycle is odd if its length is odd; otherwise, it is even.