Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T09:34:06.071Z Has data issue: false hasContentIssue false
  • English
  • Français

A Combinatorial Distinction Between Unit Circles and Straight Lines: How Many Coincidences Can they Have?

Published online by Cambridge University Press:  01 September 2009

GYÖRGY ELEKES ,
MIKLÓS SIMONOVITS  and
ENDRE SZABÓ
GYÖRGY ELEKES
Affiliation:
Mathematical Institute of Eötvös University, Hungary and Alfréd Rényi Mathematical Institute, Hungary (e-mail: [email protected], [email protected] and [email protected])
MIKLÓS SIMONOVITS
Affiliation:
Mathematical Institute of Eötvös University, Hungary and Alfréd Rényi Mathematical Institute, Hungary (e-mail: [email protected], [email protected] and [email protected])
ENDRE SZABÓ
Affiliation:
Mathematical Institute of Eötvös University, Hungary and Alfréd Rényi Mathematical Institute, Hungary (e-mail: [email protected], [email protected] and [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a very general sufficient condition for a one-parameter family of curves not to have n members with ‘too many’ (i.e., a near-quadratic number of) triple points of intersections. As a special case, a combinatorial distinction between straight lines and unit circles will be shown. (Actually, this is more than just a simple application; originally this motivated our results.)

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 5: New Directions in Algorithms, Combinatorics and Optimization , September 2009 , pp. 691 - 705
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Brass, P., Moser, W. and Pach, J. (2005) Research Problems in Discrete Geometry, Springer, New York.Google Scholar
[2]Elekes, G. (1984) n points in the plane can determine n 3/2 unit circles. Combinatorica 4 131.CrossRefGoogle Scholar
[3]Elekes, G. (1995) Circle grids and bipartite graphs of distances. Combinatorica 15 167–174.CrossRefGoogle Scholar
[4]Elekes, G. (2002) Sums versus products in number theory, algebra and Erdős geometry: A survey. In Paul Erdős and his Mathematics II, Vol. 11 of Bolyai Mathematical Society Studies, pp. 241–290.Google Scholar
[5]Elekes, G. and Szabó, E. How to find groups? (And how to use them in Erdős geometry?) Accepted in Combinatorica.Google Scholar
[6]Erdős, P. (1981) Some applications of graph theory and combinatorial methods to number theory and geometry. In Algebraic Methods in Graph Theory, Vol. 25 of Coll. Math. Soc. J. Bolyai, pp. 137–148.Google Scholar
[7]Füredi, Z. and Palásti, I. (1984) Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. 92 561–566.CrossRefGoogle Scholar
[8]Matoušek, J. (2002) Lectures on Discrete Geometry, Springer.CrossRefGoogle Scholar
[9]Pach, J. and Agarwal, P. K. (1995) Combinatorial Geometry, Wiley, New York.CrossRefGoogle Scholar
[10]Pach, J. and Sharir, M. (1990) Repeated angles in the plane and related problems. J. Combin. Theory Ser. A 59 12–22.CrossRefGoogle Scholar
[11]Spencer, J., Szemerédi, E. and Trotter, W. T. Jr, (1984, 1983) Unit distances in the Euclidean plane. In Graph Theory and Combinatorics (Cambridge 1983), Academic Press, London, pp. 293–303.Google Scholar
[12]Sylvester, J. J. (1867) Problem 2473. Math. Questions from the Educational Times 8 106–107.Google Scholar
[13]Székely, L. A. (1997) Crossing numbers and hard Erdős problems in discrete geometry. Combin. Probab. Comput. 6 353–358.CrossRefGoogle Scholar
[14]Szemerédi, E. and Trotter, W. T. Jr, (1983) Extremal problems in discrete geometry. Combinatorica 3 381–392.Google Scholar