On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry

Abstract

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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Dedicated to Paul Erdős on his seventieth birthday

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Beck, J. On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry. Combinatorica 3, 281–297 (1983). https://doi.org/10.1007/BF02579184

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AMS subject classification (1980)

  • 51 M 05
  • 05 C 35