According to Suk’s breakthrough result on the Erdős–Szekeres problem, any point set in general position in the plane, which has no n elements that form the vertex set of a convex n-gon, has at most 2n+O(n 2/3 log n) points. We strengthen this theorem in two ways. First, we show that the result generalizes to convexity structures induced by pseudoline arrangements. Second, we improve the error term. A family of n convex bodies in the plane is said to be in convex position if the convex hull of the union of no n − 1 of its members contains the remaining one. If any three members are in convex position, we say that the family is in general position. Combining our results with a theorem of Dobbins, Holmsen, and Hubard, we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes Tóth and by Pach and Tóth, respectively. Let c(n) (and c′(n)) denote the smallest positive integer N with the property that any family of N pairwise disjoint convex bodies in general position (resp., N convex bodies in general position, any pair of which share at most two boundary points) has an n-membered subfamily in convex position. We show that c(n) ≤ c′(n) ≤ 2 (√ n log n ) .

中文翻译：

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Erdős-Szekeres 问题的两个扩展

根据 Suk 在 Erdős-Szekeres 问题上的突破性结果，平面中一般位置的任何点集，如果没有 n 个元素构成凸 n 边形的顶点集，则至多有 2n+O(n 2/3记录 n) 点。我们以两种方式加强这个定理。首先，我们表明结果推广到由伪线排列引起的凸性结构。其次，我们改进了误差项。如果一个平面中的 n 个凸体的族被称为处于凸位置，如果它的非 n - 1 个成员的并集的凸包包含剩余的一个。如果任何三个成员处于凸位，我们就说这个家庭处于一般位置。将我们的结果与 Dobbins、Holmsen 和 Hubard 的定理相结合，我们显着改进了以下两个函数的最佳已知上限，分别由 Bisztriczky 和 ​​Fejes Tóth 以及 Pach 和 Tóth 介绍。令 c(n) (和 c′(n)) 表示最小的正整数 N，其性质是任何 N 个成对不相交的凸体族在一般位置（分别是 N 个一般位置的凸体，其中任何一对共享最多两个边界点）在凸位置有一个 n 成员亚族。我们证明 c(n) ≤ c′(n) ≤ 2 (√ n log n ) 。