Large bichromatic point sets admit empty monochromatic 4-gons
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We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).
CitacióAichholzer, O. [et al.]. Large bichromatic point sets admit empty monochromatic 4-gons. A: European Workshop on Computational Geometry. "25th European Workshop on Computational Geometry". 2009, p. 133-136.
Versió de l'editorhttp://2009.eurocg.org/abstracts.pdf