On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry Authors József Beck Mathematical Institute of the Hungarian Academy of Sciences 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

References [1]

J. Beck andJ. Spencer , Unit distances, submitted toJournal of Combinatorial Theory, Series A (1982)

[2]

H. S. M. Coxeter ,

Introduction to geometry , John Wiley and Sons, New York, 1961.

MATH [3]

G. A. Dirac , Collinearity properties of sets of points,

Quart. J. Math.
2 (1951) 221–227.

MATH CrossRef MathSciNet [4]

P. Erdős , On some problems of elementary and combiratorial geometry,

Annali di Mat. Pura et Applicata, Ser. IV.
103 (1975) 99–108.

CrossRef [5]

P. Erdős , Some applications of graph theory and combinatorial methods to number theory and geometry,Colloquia Math. Soc. János Bolyai, Algebraic methods in graph theory , Szeged (Hungary) (1978) 137–148.

[6]

P. Erdős , On the combinatorial problems which I would most like to see solved,

Combinatorica
1 (1981) 25–42.

MathSciNet [7]

B. Grünbaum ,Arrangements and spreads, Regional Conference Series in Mathematics
10 , Amer. Math. Soc., 1972.

[8]

E. Jucovič , Problem 24,Combinatorial structures and their applications , Gordon and Breach, New York, 1970.

[9]

L. M. Kelly and

W. Moser , On the number of ordinary lines determined by

n points,

Canad. J. Math.
10 (1958) 210–219.

MATH MathSciNet [10]

W. Moser ,Research problems in discrete geometry , Mimeograph notes, 1981.

[11]

T. S. Motzkin , The lines and planes connecting the points of a finite set,

Trans. Amer. Math. Soc.
70 (1951) 451–464.

MATH CrossRef MathSciNet [12]

E. Szemerédi and

W. T. Trotter , Extremal problems in discrete geometry,

Combinatorica
3 (3–4) (1983) 381–392.

MATH MathSciNet