We call a graph (m, k)-colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. For the class of planar graphs, and the class of outerplanar graphs, we determine all pairs (m, k) such that every graph in the class is ( m , k)-colorable. We include an elementary proof (not assuming the truth of the four-color theorem) that every planar graph is (4, 1)-colorable. Finally, we prove that, for each compact surface S, there is an integer k = k(S) such that every graph in S can be (4, &colored; we conjecture that 4 can be replaced by 3 in this statement.

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Abstract.Some useful combinatorial selection lemmas are shown to be directly equivalent to the prime ideal theorem for boolean algebras.1. The theorems we shall consider are intimately related to R. Rado's selection lemma (Theorem 2, below) which first appeared in [10] and subsequently has found wide application (see [1], [3], [4], [12]). Our main theorem is Theorem 1 which we use to derive other forms of Rado's lemma and to prove A. Robinson's valuation lemma which was shown by Robinson in [9] to be a fundamental result in model theory.We also show that Theorem 1 and some of the theorems we derive from it are equivalent to the prime ideal theorem for boolean algebras and thus we have some useful abstract versions of this theorem, versions divorced from any particular algebraic structure. Whether the original lemma of Rado is as strong as the prime ideal theorem appears to be an open question; note, however, that E. S. Wölk [12] has shown that Rado's lemma plus the axiom of choice for families of finite sets implies the Tychonoff theorem for finite spaces, which, in turn, implies the prime ideal theorem (see [8]). We conjecture that Rado's lemma is strictly weaker than the prime ideal theorem. Preliminaries.In this paper we shall work, informally, within the framework of Zermelo-Fraenkel set theory (ZF). All uses of the axiom of choice will be explicitly noted.Given a set /, by a partial function on I, we mean a function whose domain is a subset of /. A partial valuation on I is a partial function on / whose range is included in {0, 1}. As in ZF we consider functions to be special sets of ordered pairs and we even allow 0 , the empty function (the empty set of ordered pairs). If/ is a partial function on / we write D(f) for its domain, and if t/e/ we write/ft/ for the restriction of/to UnD(f).

A coin set is a strictly increasing list of positive integers that always begins with 1. A coin set is called greedy when the simple greedy change-making algorithm always produces the fewest number of coins in change. Here, the greedy change-making algorithm repeatedly selects the largest denomination coin less than the remaining amount until it has assembled the correct change. Pearson has provided an efficient algorithm for determining whether a coin set is greedy. We study a stricter property on coin sets, called total greediness, which requires that all initial subsequences of the coin set also be greedy, and a simple property makes it easy to test if a coin set is totally greedy. We begin to explore the theory of greedy obstructions– those coin sets that cannot be extended to greedy coin sets by the addition of coins in larger denominations.