A combinatorial distinction between unit circles and straight lines: How many coincidences can they have?

Authors:
GyÖrgy Elekes;MiklÓs Simonovits;Endre SzabÓ
Affiliations:
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);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);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)
Venue:
Combinatorics, Probability and Computing
Year:
2009

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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.)