Extremal problems in discrete geometry
 E. Szemerédi,
 W. T. Trotter Jr.
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In this paper, we establish several theorems involving configurations of points and lines in the Euclidean plane. Our results answer questions and settle conjectures of P. Erdõs, G. Purdy, and G. Dirac. The principal result is that there exists an absolute constantc _{1} so that when \(\sqrt n \leqq t \leqq \left( {_2^n } \right)\) , the number of incidences betweenn points andt lines is less thanc _{1} n ^{2/3} t ^{2/3}. Using this result, it follows immediately that there exists an absolute constantc _{2} so that ifk≦√n, then the number of lines containing at leastk points is less thanc _{2} n ^{2}/k ^{3}. We then prove that there exists an absolute constantc _{3} so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc _{3} n of the lines determined by then points. Finally, we show that there is an absolute constantc _{4} so that there are less than exp (c _{4} √n) sequences 2≦y _{1}≦y _{2}≦...≦y _{r} for which there is a set ofn points and a setl _{1},l _{2}, ...,l _{ t } oft lines so thatl _{ j } containsy _{ j } points.
 J. Beck, On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry,Combinatorica 3 (3–4) (1983), 281–297.
 P. Erdős, On some problems of elementary and combinatorial geometry,Annali di Mat., Ser. 4,103 (1974), 99–108. CrossRef
 P. Erdős, Some combinatorial problems in geometry,Conference held in Haifa, Israel, 1979, Lecture Notes in Mathematics,792, Springer, 1980, 46–53.
 P. Erdős, Some applications of graph theory and combinatorial methods to number theory and geometry.Algebraic methods in graph theory, Coll. Math. Soc. T. Bolyai 25 (1981), 131–148.
 P. Erdős, Problems and results in combinatorial geometry,Proceedings of New York Academy of Science, to appear.
 William Moser,Research Problems in Discrete Geometry, Mimeograph Notes (1981).
 E. Szemerédi andWilliam T. Trotter, Jr., A Combinatorial Distinction Between the Euclidean and Projective Planes,submitted.
 Title
 Extremal problems in discrete geometry
 Journal

Combinatorica
Volume 3, Issue 34 , pp 381392
 Cover Date
 19830901
 DOI
 10.1007/BF02579194
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 51 M 05
 05 C 35
 Industry Sectors
 Authors

 E. Szemerédi ^{(1)}
 W. T. Trotter Jr. ^{(2)}
 Author Affiliations

 1. Mathematical Institute of the Hungarian Academy of Sciences, Pf. 428, H1376, Budapest, Hungary
 2. Dept. of Mathematics and Statistics, University of South Carolina, 29208, Columbia, SC, USA