Abstract
Suppose S⊆ℝd is a set of (finite) cardinality n, whose complement can be written as the union of k convex sets. It is perhaps intuitively appealing that when n is large k must also be large. This is true, as is shown here. First the case in which the convex sets must also be open is considered, and in this case a family of examples yields an upper bound, while a simple application of a theorem of Björner and Kalai yields a lower bound. Much cruder estimates are then obtained when the openness restriction is dropped. For a given set S the problem of determining the smallest number of convex sets whose union is ℝd∖S is shown to be equivalent to the problem of finding the chromatic number of a certain (infinite) hypergraph ℋ S . We consider the graph \(\mathcal {G}_{S}\) whose edges are the 2-element edges of ℋ S , and we show that, when d=2, for any sufficiently large set S, the chromatic number of \(\mathcal{G}_{S}\) will be large, even though there exist arbitrarily large finite sets S for which \(\mathcal{G}_{S}\) does not contain large cliques.
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In memory of Vic Klee.
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Lawrence, J., Morris, W. Finite Sets as Complements of Finite Unions of Convex Sets. Discrete Comput Geom 42, 206–218 (2009). https://doi.org/10.1007/s00454-009-9184-0
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DOI: https://doi.org/10.1007/s00454-009-9184-0