Journal
COMBINATORICS PROBABILITY & COMPUTING
Volume 25, Issue 6, Pages 941-958Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/S0963548316000043
Keywords
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Funding
- OTKA grant [NN-102029]
- Swiss National Science Foundation [200020-144531, 200021-137574, 200020-162884]
- Israel Science Foundation [1452/15]
- United States-Israel Binational Science Foundation (BSF) [2014384]
- Fondation Sciences Mathematiques de Paris (FSMP)
- French National Research Agency (ANR) as part of the Investissements d'Avenir program [ANR-10-LABX-0098]
- EPFL, Lausanne
- Lendulet programme of the Hungarian Academy of Sciences
- Hungarian OTKA grants [NN-102029, K-116769]
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A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1 - o(1)) n(2). We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.) An important ingredient of our proofs is the following statement. Let S be a family of n open curves in R-2, so that each curve is the graph of a continuous real function defined on R, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is Omega(nt root logt/log log t).
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