A SAT attack on higher dimensional Erdős--Szekeres numbers

Abstract

A famous result by Erdős and Szekeres (1935) asserts that, for every k, d ∈ N, there is a smallest integer n = g (d) (k), such that every set of at least n points in 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) ≤ 13, g (4) (8) ≤ 13, and g (5) (9) ≤ 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 R d in general position contains a k-hole, we show h (3) (7) ≤ 14, h (4) (8) ≤ 13, and h (5) (9) ≤ 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.