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arXiv · 1207.3594

Progress on Dirac's Conjecture

Michael S. Payne·David R. Wood
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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.

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Michael S. Payne, David R. Wood. 2013-06-17. Progress on Dirac's Conjecture. https://arxiv.org/abs/1207.3594

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