Progress on Dirac’s Conjecture

In 1951, Gabriel Dirac conjectured that every non-collinear set P of n points in the plane contains a point incident to at least n2 − c of the lines determined by P, for some constant c. The following weakened conjecture was proved by Beck and by Szemerédi and Trotter: every non-collinear set P of n points in the plane contains a point in at least nc ′ lines determined by P, for some constant c ′. We prove this result with c ′ = 37. We also give the best known constant for Beck’s Theorem, proving that every set of n points with at most ` collinear determines at least 198n(n − `) lines.