Progress on Dirac's Conjecture

TL;DR

This paper advances the understanding of point-line incidences in planar point sets, providing improved lower bounds on the number of lines determined by non-collinear points, thus making progress on Dirac's conjecture.

Contribution

It proves a new lower bound of n/37 lines for non-collinear points and refines the constant in Beck's Theorem, improving previous incidence bounds.

Findings

Every set of n non-collinear points contains at least n/37 lines.

Sets with at most k collinear points determine at least n(n-k)/98 lines.

Improved constants contribute to 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 Szemer\'edi-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.