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  "path": "/abs/2603.05339v1",
  "publishedAt": "2026-03-06T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Oswin Aichholzer",
    "Helena Bergold",
    "Simon D. Fink",
    "Maarten Löffler",
    "Patrick Schnider",
    "Josef Tkadlec"
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  "textContent": "**Authors:** Oswin Aichholzer, Helena Bergold, Simon D. Fink, Maarten Löffler, Patrick Schnider, Josef Tkadlec\n\nWe 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"
}