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Ramsey-type results for semi-algebraic relations

Authors :
János Pach
David Conlon
Andrew Suk
Benny Sudakov
Jacob Fox
Massachusetts Institute of Technology. Department of Mathematics
Fox, Jacob
Suk, Andrew
Source :
Scopus-Elsevier, arXiv, Symposium on Computational Geometry
Publication Year :
2014
Publisher :
American Mathematical Society (AMS), 2014.

Abstract

For natural numbers d and t there exists a positive C such that if F is a family of n[superscript C] semi-algebraic sets in R[superscript d] of description complexity at most t, then there is a subset F' of F of size $n$ such that either every pair of elements in F' intersect or the elements of F' are pairwise disjoint. This result, which also holds if the intersection relation is replaced by any semi-algebraic relation of bounded description complexity, was proved by Alon, Pach, Pinchasi, Radoicic, and Sharir and improves on a bound of 4[superscript n] for the family F which follows from a straightforward application of Ramsey's theorem. We extend this semi-algebraic version of Ramsey's theorem to k-ary relations and give matching upper and lower bounds for the corresponding Ramsey function, showing that it grows as a tower of height k-1. This improves on a direct application of Ramsey's theorem by one exponential. We apply this result to obtain new estimates for some geometric Ramsey-type problems relating to order types and one-sided sets of hyperplanes. We also study the off-diagonal case, achieving some partial results.<br />Simons Foundation (Fellowship)<br />National Science Foundation (U.S.) (Grant DMS 1069197)<br />NEC Corporation (MIT Award)<br />National Science Foundation (U.S.) (Postdoctoral Fellowship)<br />Swiss National Science Foundation (Grant 200021-125287/1)

Details

ISSN :
10886850 and 00029947
Volume :
366
Database :
OpenAIRE
Journal :
Transactions of the American Mathematical Society
Accession number :
edsair.doi.dedup.....4614090021570c27ae72abd078c578ae
Full Text :
https://doi.org/10.1090/s0002-9947-2014-06179-5