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Notes on the 33-point Erdős--Szekeres problem

Bogdan Dumitru

TL;DR

The paper tackles the open planar Erdős–Szekeres problem for $ES(7)$ by encoding point configurations with triple-orientation Booleans and a 4-set convexity reduction to exclude convex 7-gons. It combines CC-style 5-point implications, a 4-set pattern selector framework, and convex-layer anchoring (hull templates) plus a simple sub-cubing parameter to generate structured subinstances. The approach yields UNSAT certificates for several anchored subfamilies, though runtime varies widely across configurations, sometimes dramatically. The work highlights practical encoding challenges and proposes concrete directions—coarsening 4-set types and tightening outer-layer anchoring—for improved scalability and robustness in future SAT-based explorations of ES(7).

Abstract

The determination of $ES(7)$ is the first open case of the planar Erdős--Szekeres problem, where the general conjecture predicts $ES(7)=33$. We present a SAT encoding for the 33-point case based on triple-orientation variables and a 4-set convexity criterion for excluding convex 7-gons, together with convex-layer anchoring constraints. The framework yields UNSAT certificates for a collection of anchored subfamilies. We also report pronounced runtime variability across configurations, including heavy-tailed behavior that currently dominates the computational effort and motivates further encoding refinements.

Notes on the 33-point Erdős--Szekeres problem

TL;DR

The paper tackles the open planar Erdős–Szekeres problem for by encoding point configurations with triple-orientation Booleans and a 4-set convexity reduction to exclude convex 7-gons. It combines CC-style 5-point implications, a 4-set pattern selector framework, and convex-layer anchoring (hull templates) plus a simple sub-cubing parameter to generate structured subinstances. The approach yields UNSAT certificates for several anchored subfamilies, though runtime varies widely across configurations, sometimes dramatically. The work highlights practical encoding challenges and proposes concrete directions—coarsening 4-set types and tightening outer-layer anchoring—for improved scalability and robustness in future SAT-based explorations of ES(7).

Abstract

The determination of is the first open case of the planar Erdős--Szekeres problem, where the general conjecture predicts . We present a SAT encoding for the 33-point case based on triple-orientation variables and a 4-set convexity criterion for excluding convex 7-gons, together with convex-layer anchoring constraints. The framework yields UNSAT certificates for a collection of anchored subfamilies. We also report pronounced runtime variability across configurations, including heavy-tailed behavior that currently dominates the computational effort and motivates further encoding refinements.
Paper Structure (26 sections, 1 theorem, 15 equations, 1 table)

This paper contains 26 sections, 1 theorem, 15 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: triple orientations and CC-style constraints
  3. Orientation variables
  4. A 5-point implication (CC axiom 5)
  5. Reduced generation.
  6. Interpretation of solver outcomes.
  7. A 4-set criterion for convex position
  8. Encoding 4-point types and excluding convex 7-sets
  9. Four cyclic triple orientations
  10. Selector variables and CNF
  11. No convex 7-set clauses
  12. Convex layers (hull templates) and a simple sub-cubing parameter
  13. Convex-layer (hull template) anchoring
  14. A simple sub-cubing parameter (anchoring consecutive layers)
  15. Geometric meaning.
  16. ...and 11 more sections

Key Result

Proposition 1

Let $S$ be a finite set of points in the plane in general position. Then $S$ is in convex position if and only if every 4-point subset of $S$ is in convex position.

Theorems & Definitions (5)

  • Definition 1: 3-cup / 3-cap (convention)
  • Proposition 1: 4-set criterion for convex position
  • proof : Proof sketch
  • Remark 1: Convex vs. non-convex patterns
  • Remark 2: Runtime variability