Abstract

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

Citations

136	Combinatorial complexity bounds for arrangements of curves and spheres, Discrete and Computational Geometry 5 - Clarkson, Edelsbrunner, et al. - 1990
99	On sets of distances of n points - Erdős - 1970
63	Crossing-free subgraphs - Ajtai, Chvátal, et al.
60	Crossing number problems - Erdős, Guy - 1973
32	On the lattice property of the plane and some problems of Dirac - Beck - 1983
17	The number of different distances determined by n points in the plane - Chung - 1984
17	Crossing number of graphs - Guy - 1972
16	The number of different distances determined by a set of points in the Euclidean plane, Discrete - Chung, Szemerédi, et al. - 1992
12	Problems and results in combinatorial geometry - Erdos - 1985
8	Traces of finite sets: Extremal problems and geometric applications - Füredi, Pach - 1994
5	Unit distances - Beck, Spencer
4	On the number of dierent distances determined by n points in the plane - Chung
4	The number of dierent distances determined by a set of points in the Euclidean plane, Discrete Comp. Geometry 7 - Chung, Szemeredi, et al. - 1992
1	Traces of nite sets: extremal problems and geometric applications, in: Extremal Problems for Finite Sets, Visegrad - Furedi, Pach - 1991