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Garment numbers of bi-colored point sets in the plane

Oswin Aichholzer, Helena Bergold, Simon D. Fink, Maarten Löffler, Patrick Schnider, Josef Tkadlec

Abstract

We consider colored variants of a class of geometric-combinatorial questions on $k$-gons and empty $k$-gons that have been started around 1935 by Erdős and Szekeres. In our setting we have $n$ points in general position in the plane, each one colored either red or blue. A structure on $k$ points is a geometric graph where the edges are spanned by (some of) these points and is called monochromatic if all $k$ points have the same color. Already for $k=4$ there exist interesting open problems. Most prominently, it is still open whether for any sufficiently large bichromatic set there always exists a convex empty, monochromatic quadrilateral. In order to shed more light on the underlying geometry we study the existence of five different monochromatic structures that all use exactly 4 points of a bichromatic point set. We provide several improved lower and upper bounds on the smallest $n$ such that every bichromatic set of at least $n$ points contains (some of) those monochromatic structures.

Garment numbers of bi-colored point sets in the plane

Abstract

We consider colored variants of a class of geometric-combinatorial questions on -gons and empty -gons that have been started around 1935 by Erdős and Szekeres. In our setting we have points in general position in the plane, each one colored either red or blue. A structure on points is a geometric graph where the edges are spanned by (some of) these points and is called monochromatic if all points have the same color. Already for there exist interesting open problems. Most prominently, it is still open whether for any sufficiently large bichromatic set there always exists a convex empty, monochromatic quadrilateral. In order to shed more light on the underlying geometry we study the existence of five different monochromatic structures that all use exactly 4 points of a bichromatic point set. We provide several improved lower and upper bounds on the smallest such that every bichromatic set of at least points contains (some of) those monochromatic structures.
Paper Structure (7 sections, 3 theorems, 5 figures, 1 table)

This paper contains 7 sections, 3 theorems, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper Bounds
  4. Lower bounds
  5. Conclusion
  6. Upper Bounds with Pants
  7. Upper Bound for only Necklace

Key Result

theorem 1

We have $\mathcal{G}(\mathop{\mathrm{pant}}\nolimits\vee \mathop{\mathrm{necklace}}\nolimits)\le 21$.

Figures (5)

  • Figure 1: The five types of structures: cravat, necklace, bowtie, skirt, and pant.
  • Figure 2: Left: A set of 7 points, 5 of them red and 2 blue. Middle: Each monochromatic skirt and bowtie contains a point of the other color, thus we have $\mathcal{G}(\mathop{\mathrm{skirt}}\nolimits\vee \,\mathrm{bowtie})>7$. Right: There is an empty pant, so the point set does not imply anything about the setting $\mathop{\mathrm{pant}}\nolimits\vee\,\mathrm{bowtie}$.
  • Figure 3: The two cases of \ref{['lem:induction']}.
  • Figure 4: Left: We have $\textcolor{red}{ 4}\xspace \triangleright \textcolor{blue}{2}\xspace$, regardless of whether the red quadrilateral is convex or not. Right: We have $\textcolor{red}{5}\xspace \triangleright \textcolor{blue}{3}\xspace$, regardless of the size of the convex hull of the red points.
  • Figure 5: Our lower bound constructions, all avoiding certain empty monochromatic structures.

Theorems & Definitions (14)

  • proof : Proof sketch.
  • theorem 1
  • proof
  • theorem 2
  • proof
  • proof : Proof sketch.
  • proof
  • proof
  • proof
  • proof
  • ...and 4 more