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

Received: 11 September 1982 DOI :
10.1007/BF02579184

Cite this article as: Beck, J. Combinatorica (1983) 3: 281. doi:10.1007/BF02579184 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 .

AMS subject classification (1980) 51 M 05 05 C 35 Dedicated to Paul Erdős on his seventieth birthday

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MATH MathSciNet Authors and Affiliations 1. Mathematical Institute of the Hungarian Academy of Sciences Budapest Hungary