Erdős-Szekeres and Testing Weak epsilon-Nets are NP-hard in 3 dimensions - and what now?

On the Erdos-Szekeres convex polygon problem

Let $ES(n)$ be the smallest integer such that any set of $ES(n)$ points in the plane in general position contains $n$ points in convex position. In their seminal 1935 paper, Erdos and Szekeres showed that $ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}$. In 1960, they showed that $ES(n) \geq 2^{n-2} + 1$ and conjectured this to be optimal. In this paper, we nearly settle the Erdos-Szekeres conjecture by showing that $ES(n) =2^{n +o(n)}$.

关于33点埃罗什-塞克雷斯问题的笔记

Notes on the 33-point Erdős--Szekeres problem

本文提出了一种基于三向量变量和四集凸性准则的SAT编码，用于解决33点埃罗什-塞克雷斯问题，并探讨了计算复杂性和编码优化的挑战。

Two player game variant of the Erdos-Szekeres problem

The classical Erdos-Szekeres theorem states that a convex $k$-gon exists in every sufficiently large point set. This problem has been well studied and finding tight asymptotic bounds is considered a challenging open problem. Several variants of the Erdos-Szekeres problem have been posed and studied in the last two decades. The well studied variants include the empty convex $k$-gon problem, convex $k$-gon with specified number of interior points and the chromatic variant. In this paper, we introduce the following two player game variant of the Erdos-Szekeres problem: Consider a two player game …

A SAT attack on higher dimensional Erdős--Szekeres numbers

A famous result by Erdős and Szekeres (1935) asserts that, for all $k,d \in \mathbb{N}$, there is a smallest integer $n = g^{(d)}(k)$ such that every set of at least $n$ points in $\mathbb{R}^d$ in general position contains a $k$-gon, that is, a subset of $k$ points which is in convex position. In this article, we present a SAT model based on acyclic chirotopes (oriented matroids) to investigate Erdős--Szekeres numbers in small dimensions. To solve the SAT instances we use modern SAT solvers and all our unsatisfiability results are verified using DRAT certificates. We show $g^{(3)}(7) = 13$, $…

Erdős--Szekeres-type problems in the real projective plane

We consider point sets in the real projective plane $\mathbb{R}P^2$ and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős--Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erdős--Szekeres theorem about point sets in convex position in $\mathbb{R}P^2$, which was initiated by Harborth and Möller in 1994. The notion of convex position in $\mathbb{R}P^2$ agrees with the definition of convex sets introduced by Steinitz in 1913. For $k \geq 3$, an (\affine) $k$-hole in a finite set $S \subseteq \math…

Optimality program in segment and string graphs

Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{Θ(n)}$ is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among …