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This paper introduces a new approach to the T-coloring problem for complete graphs. The problem arises from Hale's formulation of the channel assignment problem for potentially interfering communication nets. The motivating result of this paper is that the T-span of Kn, denoted sp-(Kn), is asymptotically independent of n. More precisely, each T-set has a rate, rt (T), and n/spT (K,) converges to rt (T). We introduce a finite algorithm for computing the rate of T. This is accomplished by associating to a given set T an infinite sequence of integers with the property that the first n integers of this sequence T-color K, in an asymptotically optimal way. Lastly, we compute rt (T) or bounds on its value for some interesting special cases of sets T.

Abstract. Let X be a reduced compact complex space, 50~X a coherent sheaf, and V = V(50) its associated linear fiber space. Let V R be the reduction of V, let A be the analytic set in X over which 50 is not locally-free, and let V' be the closure in V R of VRI (X--A). 6 e is (primary) weakly positive if the zerosection of V (V') is exceptional. 50 is (primary) cohomologically positive if, for any coherent sheaf ~-~ X,for all g >> 0, k > t. Then 50 is (primary) weakly positive if and only if 50 is (primary) cohomologically positive.Let X be a normal irreducible compact complex space. Then X is Moishezon if and only if it carries a primary weakly positive, and hence primary cohomologically positive, coherent sheaf.Several other positivity notions are also discussed.The author wishes to express his sincere gratitude to Professor Masatake Kuranishi for his generous advice and support during the last five years.He would also like to thank Professor Raghavan Narasimhan for several helpful discussions during his work on these results. He is particularly indebted to Professor Narasimhan for reading the entire manuscript.1. Let X be a reduced compact complex space and 50-~ X a coherent analytic sheaf. Let V---V(50) be the linear fiber space dual to 5 ~ (see [5,2]), let V R be the reduction of V and let A be the analytic set in X over which 50 is not locally-free (see [-20]). The primary component of V(50), denoted V'(50), is the closure in V R of VRI (X--A). Note that, in general, V R and V' are not linear fiber spaces; they are, however, invariant under the action of C*. Let P=P (50) By a metric in V, I mean a metric in the sense of Grauert-Riemenschneider (see [7,18]). That is, a metric in V consists of hermitian forms {hx}x~ x on the

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