Consider a graph obtained by taking an edge disjoint union of k complete bipartite graphs. Alon, Saks, and Seymour conjectured that such graphs have chromatic number at most k+1. This well known conjecture remained open for almost twenty years. In this paper, we construct a counterexample to this conjecture and discuss several related problems in combinatorial geometry and communication complexity.
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∗ Research supported in part by NSF CAREER award DMS-0812005 and by a USA-Israeli BSF grant. Research supported in part by NSF grant DMS-1101185, NSF CAREER award DMS-0812005, and by a USA-Israeli BSF grant.