A weak reduction of the Erd\"os-Szekeres conjecture into a constraint unsatisfiability problem regarding certain multisets

TL;DR

This paper introduces div point sets to analyze point configurations and reduces the Erd"os-Szekeres conjecture to a first-order logic unsatisfiability problem involving multisets, offering a new approach to this longstanding combinatorial problem.

Contribution

It presents a novel framework called div point sets and links the Erd"os-Szekeres conjecture to logical unsatisfiability, providing a new perspective on the problem.

Findings

Proposes div point sets as a framework for point set analysis.

Reduces the Erd"os-Szekeres conjecture to a logical unsatisfiability problem.

Suggests a potential pathway to prove the conjecture via logic.

Abstract

We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order logic formulae concerning some sets of multisets of uniform cardinality over boolean variables would prove the Erd\"os-Szekeres conjecture, which states that for any set of 2^(n-2)+1 points in general position, there exists n points forming a convex polygon, where n is greater than or equal to 3.