Two extensions of the Erdős–Szekeres problem

Abstract

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 2 n+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+O(√ n log n) .

A new discrete theory of pseudoconvexity

“…Further, it is interesting to see if problems open about convex sets can be improved in the context of pseudoconvex sets, in particular finding a maximal subset of points in a convex position, bounding the size of weak epsilon-nets for convex sets or the number of k-sets. We mentioned earlier that the best known result about the size of a maximal convex subset of points does extend to the geometric pseudohalfplane setting via TAPs Holmsen et al (2022) and thus also to pseudohalfplanes. It would be interesting to give a direct proof of the respective statement about pseudohalfplane hypergraphs using our methods.…”

“…One can check that two-coloring the triples according to this to get color classes T cup and T cap , we get a transitive coloring, i.e., for which (s 1 , s 2 , s 3 ), (s 2 , s 3 , s 4 ) ∈ T i ⇒ (s 1 , s 2 , s 4 ), (s 1 , s 3 , s 4 ) ∈ T i whenever s 1 < s 2 < s 3 < s 4 , i ∈ {cup, cap}. (xiii) We can conclude the proof using the version of the Erdős-Szekeres Cup-Cap Theorem for transitive colorings (see Fox et al (2012); Hubard et al (2011); Moshkovitz and Shapira (2014); Holmsen et al (2022)).…”

“…However, it is quite straightforward to check how the conditions and conclusions of our theorems change when adding new hyperedges to the pseudohalfplane hypergraph, and see that every one of these results on a H pseudohalfplane hypergraph implies the same on its subhypergraph H. Thus, earlier results about TAPs do imply our respective results about pseudohalfplane families and in turn also about pseudohalfplane hypergraphs. Furthermore, in Holmsen et al (2022) they consider the Erdős-Szekeres problem in TAPs and prove a result which is best known even in the plane. In addition, a mostly combinatorial proof using wiring diagrams of the Cup-Cap Theorem for finite families of pseudohalfplanes (i.e.…”

Counting Non-Convex 5-Holes in a Planar Point Set

“…Suk [18] in 2017 made a breakthrough N(k) ≤ 2 k+O(k 2/3 log k) . Currently, the best upper bound is N(k) ≤ 2 k+O( √ k log k) due to Holmsen et al [19].…”

A SAT attack on higher dimensional Erdős--Szekeres numbers

“…There were several improvements of the upper bound in the past decades, each of magnitude 4 k−o(k) , and in 2016, Suk showed g (2) (k) ≤ 2 k+o(k) [Suk17]. Shortly after, Holmsen et al [HMPT20] slightly improved the error term in the exponent and presented a generalizion to chirotopes (see Section 2 for a definition). The lower bound g (2) (k) ≥ 2 k−2 + 1 is known to be sharp for k ≤ 6.…”