On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry
- József Beck
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LetS be a set ofn non-collinear points in the Euclidean plane. It will be shown here that for some point ofS the number ofconnecting lines through it exceedsc · n. This gives a partial solution to an old problem of Dirac and Motzkin. We also prove the following conjecture of Erdős: If any straight line contains at mostn−x points ofS, then the number of connecting lines determined byS is greater thanc · x · n.
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- On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry
Volume 3, Issue 3-4 , pp 281-297
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- József Beck (1)
- Author Affiliations
- 1. Mathematical Institute of the Hungarian Academy of Sciences, Pf. 428, H—1395, Budapest, Hungary