On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry

Abstract

Let S be a set of n non-collinear points in the Euclidean plane. It will be shown here that for some point of S the number of connecting lines through it exceeds c · n. This gives a partial solution to an old problem of Dirac and Motzkin. We also prove the following conjecture of Erdo{combining double acute accent}s: If any straight line contains at most n-x points of S, then the number of connecting lines determined by S is greater than c · x · n. © 1983 Akadémiai Kiadó.