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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: elekes@cs.elte.hu, miki@renyi.hu and endre@renyi.hu)
MIKLÓS SIMONOVITS
Affiliation:
Mathematical Institute of Eötvös University, Hungary and Alfréd Rényi Mathematical Institute, Hungary (e-mail: elekes@cs.elte.hu, miki@renyi.hu and endre@renyi.hu)
ENDRE SZABÓ
Affiliation:
Mathematical Institute of Eötvös University, Hungary and Alfréd Rényi Mathematical Institute, Hungary (e-mail: elekes@cs.elte.hu, miki@renyi.hu and endre@renyi.hu)
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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.)

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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

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