Notes on the 33-point Erd\H{o}s--Szekeres problem

Bogdan Dumitru

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arXiv · 2512.24061

Notes on the 33-point Erd\H{o}s--Szekeres problem

Bogdan Dumitru

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Abstract

The determination of $ES(7)$ is the first open case of the planar Erd\H{o}s--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.

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Bogdan Dumitru. 2025-12-30. Notes on the 33-point Erd\H{o}s--Szekeres problem. https://arxiv.org/abs/2512.24061

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