A SAT Attack on Higher Dimensional Erdős–Szekeres Numbers

Abstract

A famous result by Erdős and Szekeres (1935) asserts that, for every \(k,d \in \mathbb {N}\), 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, i.e., a subset of k points which is in convex position. We present a SAT model for higher dimensional point sets which is based on chirotopes, and use modern SAT solvers to investigate Erdős–Szekeres numbers in dimensions \(d=3,4,5\). We show \(g^{(3)}(7) \le 13\), \(g^{(4)}(8) \le 13\), and \(g^{(5)}(9) \le 13\), which are the first improvements for decades. For the setting of k-holes (i.e., k-gons with no other points in the convex hull), where \(h^{(d)}(k)\) denotes the minimum number n such that every set of at least n points in \(\mathbb {R}^d\) in general position contains a k-hole, we show \(h^{(3)}(7) \le 14\), \(h^{(4)}(8) \le 13\), and \(h^{(5)}(9) \le 13\). Moreover, all obtained bounds are sharp in the setting of chirotopes and we conjecture them to be sharp also in the original setting of point sets.