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Intersection patterns of curves
Author(s) -
Fox Jacob,
Pach János,
Tóth Csaba D.
Publication year - 2011
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdq087
Subject(s) - intersection (aeronautics) , disjoint sets , mathematics , combinatorics , plane (geometry) , family of curves , constant (computer programming) , property (philosophy) , geometry , computer science , philosophy , epistemology , engineering , programming language , aerospace engineering
We prove that for every k ∈ ℕ there is a constant c k > 0 with the following property. Every set of n > 1 continuous curves in the plane, any pair of which intersect in at most k points, has two disjoint subsets A and B , each of size at least c k n , such that either every curve in A intersects all curves in B , or no curve in A intersects any curve in B . This statement does not remain true if we drop the condition on the number of intersection points per pair.