Ramsey-type results for semi-algebraic relations
Ramsey-type results for semi-algebraic relations
复制标题
半代数关系的 Ramsey 型结果
DOI：
10.1145/2462356.2462399
复制
发表时间：
2013
影响因子：
1
通讯作者：
Andrew Suk
中科院分区：
文献类型：
--
作者：
D. Conlon;J. Fox;J. Pach;B. Sudakov;Andrew Suk
对于自然数字D和T，存在一个正c，使得F是NC半代数集的家族，最多最多是描述复杂性的RD，则有一个大小$ n $的子集F'f'f'f'f' f'相交或f'元素中的每对元素都是成对的脱节。如果交点关系被任何半代数关系取代有界描述复杂性的任何半代数关系，则该结果也得出拉姆齐定理的直接应用。我们将Ramsey定理的该半代数版本扩展到K-Ary关系，并为相应的Ramsey函数提供匹配的上和下限，表明它随着高度K-1的塔而生长。这可以通过一个指数来改善Ramsey定理的直接应用。我们将此结果应用于获得与订单类型和单方面超平面相关的一些几何拉姆齐型问题的新估计。我们还研究了非对角线病例，从而取得了一些部分结果。
For natural numbers d and t there exists a positive C such that if F is a family of nC semi-algebraic sets in Rd 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 4n 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.