Progress on Dirac's Conjecture

Abstract

In 1951, Gabriel Dirac conjectured that every set P of n non-collinear points in the plane contains a point in at least n/2-c lines determined by P, for some constant c. The following weakening was proved by Beck and Szemerédi-Trotter: every set P of n non-collinear points contains a point in at least n/c lines determined by P, for some large unspecified constant c. We prove that every set P of n non-collinear points contains a point in at least n/37 lines determined by P. We also give the best known constant for Beck's Theorem, proving that every set of n points with at most k collinear determines at least n(n-k)/98 lines.

arXiv comment8 pages, 1 figure. Version 3 improves constant in main result via use of Hirzebruch's inequality, and adds section on Beck's theorem. Version 4 fixes formatting errors in html abstract (pdf unchanged)