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

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Abstract

Let S be a set of n non-collinear points in the Euclidean plane. It will be shown here that for some point of S the number of connecting lines through it exceeds c · n. This gives a partial solution to an old problem of Dirac and Motzkin. We also prove the following conjecture of Erdo{combining double acute accent}s: If any straight line contains at most n-x points of S, then the number of connecting lines determined by S is greater than c · x · n.

Original languageEnglish (US)
Pages (from-to)281-297
Number of pages17
JournalCombinatorica
Volume3
Issue number3-4
DOIs
StatePublished - Sep 1983

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

Keywords

  • AMS subject classification (1980): 51M05, 05C35

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