Garment numbers of bi-colored point sets in the plane
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910042533396480 |
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| author | Aichholzer, Oswin Bergold, Helena Fink, Simon D. Löffler, Maarten Schnider, Patrick Tkadlec, Josef |
| author_facet | Aichholzer, Oswin Bergold, Helena Fink, Simon D. Löffler, Maarten Schnider, Patrick Tkadlec, Josef |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05339 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Garment numbers of bi-colored point sets in the plane Aichholzer, Oswin Bergold, Helena Fink, Simon D. Löffler, Maarten Schnider, Patrick Tkadlec, Josef Computational Geometry 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. |
| title | Garment numbers of bi-colored point sets in the plane |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2603.05339 |