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.

Abstract not found

problems whose proposers believe them to be within reach of existing methods. Manuscripts should preferably contain the background of the problem and all references known to the author. The length of the manuscripts should not exceed two type-written pages. 36. Let Xn = {x,,..., x„} be n points in the plane. I will say that the set X,; has property Pk if no line contains more than k of the points. Thus property P„ _ i means that not all the points are on a line. I stated many conjectures on the number of lines determined by the set X,,. Many of my conjectures have recently been proved by Beck, F. Chung, Spencer, Szemerédi, Trotter and others [1], [2], [11]. But one of my old conjectures remained open. Let X „ have property P,., k j 3. Denote by /k (n) the maximal number of lines which contain k points of X, I conjectured that for fixed k if n (1) fk(n')