@article{de06e5e6037148b3a949841a2595c834,
title = "Planar point sets determine many pairwise crossing segments",
abstract = "We show that any set of n points in general position in the plane determines n1−o(1) pairwise crossing segments. The best previously known lower bound, Ω(n), was proved more than 25 years ago by Aronov, Erd{\H o}s, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.",
keywords = "Avoiding sets, Crossing families, Epsilon-nets, Geometric graphs, Intersection graphs, Partial orders",
author = "J{\'a}nos Pach and Natan Rubin and G{\'a}bor Tardos",
note = "Funding Information: Supported by NKFIH grants K-131529, KKP-133864, Austrian Science Fund Z 342-N31, Ministry of Education and Science of the Russian Federation MegaGrant No. 075-15-2019-1926, ERC Advanced Grant “GeoScape.”Ralph Selig Career Development Chair in Information Theory. The project leading to this application has received funding from European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No. 678765. Also supported by grant 1452/15 from Israel Science Foundation.Supported by the “Lend{\"u}let” Project in Cryptography of the Hungarian Academy of Sciences, the National Research, Development and Innovation Office NKFIH projects K-116769, SNN-117879, K-132696, KKP-133864 and SNN-13564, the ERC Advanced Grant “GeoScape” and by the Ministry of Education and Science of the Russian Federation MegaGrant No. 075-15-2019-1926. Publisher Copyright: {\textcopyright} 2021",
year = "2021",
month = aug,
day = "6",
doi = "10.1016/j.aim.2021.107779",
language = "English",
volume = "386",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",
}