A SAT Attack on Erdős-Szekeres Numbers in R^d and the Empty Hexagon Theorem

Abstract

 A famous result by Erdős and Szekeres (1935) asserts that, for all k, d in 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, that is, a subset of k points in convex position.
We present a SAT model based on acyclic chirotopes (oriented matroids) to investigate Erdős-Szekeres numbers in small dimensions. SAT instances are solved using modern SAT solvers and unsatisfiability results are verified using DRAT certificates.
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 smallest integer 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. All obtained bounds are sharp in the setting of acyclic chirotopes and we conjecture them to be sharp also in the original setting of point sets.
Last but not least, we verify all previously known bounds and, in particular, we present the first computer-assisted proof for the existence of 6-holes in sufficiently planar point sets by verifying Gerken's estimate h⁽²⁾(6) ≤ g⁽²⁾(9).