The Crossing Lemma Revisited

The Crossing Lemma  of Ajtai, Chvatal, Newborn, Szemeredi (1982) and Leighton (1983)states that if a graph of n vertices and e>4n edges is drawn in the plane, then the number of crossings between its edges must be at least constant times e^3/n^2. This statement, which is asymptotically tight, has found many applications in combinatorial geometry and in additive combinatorics. However, most results obtained using the Crossing Lemma do not appear to be optimal, and there is a quest for improved versions of the lemma for graphs satisfying certain special properties. In this talk, I describe some recent extensions of the lemma to multigraphs (joint work with G. Toth) and to families of continuous arcs in the plane (joint work with N. Rubin and G. Tardos, and with G. Tardos).

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