Crossing Numbers and Hard Erd} os Problems in Discrete Geometry

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 amongn points, the minimum number of distinct distances among n points. \A statement about curves is not interesting unless it is already interesting in the case of a circle." (H. Steinhaus)

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