Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons

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 Ê2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).

• Type: Journal
• Year: 2010
• Where: SIAMDM
• Authors: Oswin Aichholzer, Thomas Hackl, Clemens Huemer, Ferran Hurtado, Birgit Vogtenhuber
• Keywords: Convex K-gon, Empty, General Position, SIAMDM 2010