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Saturation Results Around the Erdős-Szekeres Problem

Authors Gábor Damásdi , Zichao Dong, Manfred Scheucher , Ji Zeng

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Author Details

Gábor Damásdi
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • ELTE Eötvös Loránd University, Budapest, Hungary
Zichao Dong
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, South Korea
Manfred Scheucher
  • Institut für Mathematik, Technische Universität Berlin, Germany
Ji Zeng
  • University of California San Diego, La Jolla, CA, USA
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary

Funding

  • Damásdi, Gábor: Research partially supported by ERC grant No. 882971, "GeoScape".
  • Dong, Zichao: Research supported by ERC grant No. 882971, "GeoScape", by the Erdős Center, and by the Institute for Basic Science (IBS-R029-C4).
  • Scheucher, Manfred: Supported by DFG grant SCHE 2214/1-1.
  • Zeng, Ji: Research supported by NSF grant DMS-1800746, ERC grant No. 882971, "GeoScape", and by the Erdős Center.

Cite AsGet BibTex

Gábor Damásdi, Zichao Dong, Manfred Scheucher, and Ji Zeng. Saturation Results Around the Erdős-Szekeres Problem. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.46

Abstract

In this paper, we consider saturation problems related to the celebrated Erdős-Szekeres convex polygon problem. For each n ≥ 7, we construct a planar point set of size (7/8) ⋅ 2^{n-2} which is saturated for convex n-gons. That is, the set contains no n points in convex position while the addition of any new point creates such a configuration. This demonstrates that the saturation number is smaller than the Ramsey number for the Erdős-Szekeres problem. The proof also shows that the original Erdős-Szekeres construction is indeed saturated. Our construction is based on a similar improvement for the saturation version of the cups-versus-caps theorem. Moreover, we consider the generalization of the cups-versus-caps theorem to monotone paths in ordered hypergraphs. In contrast to the geometric setting, we show that this abstract saturation number is always equal to the corresponding Ramsey number.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Convex polygon
  • Cups-versus-caps
  • Monotone path
  • Saturation problem

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References

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