# Dirac's Conjecture

**Conjecture**For every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some constant .

In 1983, Beck[B], and independently Szemerédi and Trotter [ST], proved that for every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some large unspecified constant . Payne and Wood [PW] proved this result with .

## Bibliography

[B] Jozsef Beck On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry. Combinatorica, 3(3-4):281–297, 1983.

*[D] Gabriel A. Dirac. Collinearity properties of sets of points. Quart. J. Math., Oxford Ser. (2), 2:221–227, 1951. MR: 0043485.

[PW] Michael S. Payne and David R. Wood. Progress on Dirac's Conjecture. Electronic J. Combinatorics 21.2:P2.12, 2014. arXiv:1207.3594.

[ST] Endre Szemerédi and William T. Trotter, Jr., Extremal problems in discrete geometry. Combinatorica 3.3-5:381-392, 1983.

* indicates original appearance(s) of problem.