Progress on Dirac's Conjecture

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 ` collinear determines at least 1 98 n(n− `) lines.