@article{7f76e33478034ddcb827e98c1b43e421,
title = "Crossing patterns of semi-algebraic sets",
abstract = "We prove that, for every family F of n semi-algebraic sets in ℝd of constant description complexity, there exist a positive constant ε that depends on the maximum complexity of the elements of F, and two subfamilies F1, F2 ⊆ F with at least εn elements each, such that either every element of F1 intersects all elements of F2 or no element of F1 intersects any element of F2. This implies the existence of another constant δ such that F has a subset F′ ⊆ F with nδ elements, so that either every pair of elements of F′ intersect each other or the elements of F′ are pairwise disjoint. The same results hold when the intersection relation is replaced by any other semi-algebraic relation. We apply these results to settle several problems in discrete geometry and in Ramsey theory.",
keywords = "Borsuk-Ulam theorem, Crossing patterns, Ramsey theory, Range searching, Real algebraic geometry",
author = "Noga Alon and J{\'a}nos Pach and Rom Pinchasi and Rado{\v s} Radoi{\v c}i{\'c} and Micha Sharir",
note = "Funding Information: J{\'a}nos Pach and Micha Sharir have been supported by NSF Grant CCR-00-98246, and by a grant from the US-Israeli Binational Science Foundation. Work by J{\'a}nos Pach has also been supported by PSC-CUNY Research Award 63352-0036, and by OTKA T-032458. Noga Alon has been supported by a grant from the US-Israeli Binational Science Foundation, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Work by Micha Sharir has also been supported by a grant from the Israel Science Fund, Israeli Academy of Sciences, for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.",
year = "2005",
month = aug,
doi = "10.1016/j.jcta.2004.12.008",
language = "אנגלית",
volume = "111",
pages = "310--326",
journal = "Journal of Combinatorial Theory. Series A",
issn = "0097-3165",
publisher = "Academic Press Inc.",
number = "2",
}