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.
Similar content being viewed by others
References
J. Beck andJ. Spencer, Unit distances, submitted toJournal of Combinatorial Theory, Series A (1982)
H. S. M. Coxeter,Introduction to geometry, John Wiley and Sons, New York, 1961.
G. A. Dirac, Collinearity properties of sets of points,Quart. J. Math. 2 (1951) 221–227.
P. Erdős, On some problems of elementary and combiratorial geometry,Annali di Mat. Pura et Applicata, Ser. IV. 103 (1975) 99–108.
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.
P. Erdős, On the combinatorial problems which I would most like to see solved,Combinatorica 1 (1981) 25–42.
B. Grünbaum,Arrangements and spreads, Regional Conference Series in Mathematics 10, Amer. Math. Soc., 1972.
E. Jucovič, Problem 24,Combinatorial structures and their applications, Gordon and Breach, New York, 1970.
L. M. Kelly andW. Moser, On the number of ordinary lines determined byn points,Canad. J. Math. 10 (1958) 210–219.
W. Moser,Research problems in discrete geometry, Mimeograph notes, 1981.
T. S. Motzkin, The lines and planes connecting the points of a finite set,Trans. Amer. Math. Soc. 70 (1951) 451–464.
E. Szemerédi andW. T. Trotter, Extremal problems in discrete geometry,Combinatorica 3 (3–4) (1983) 381–392.
Author information
Authors and Affiliations
Additional information
Dedicated to Paul Erdős on his seventieth birthday
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02579184