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 talk, we will show a counterexample to this conjecture. this construction will also lead to some related results in combinatorial geometry and communication complexity. in particular, it implies a nontrivial lower bound of the non-deterministic communication complexity of the ``clique versus independent set'' problem.\\ joint work with benny sudakov.