Home » Subject » Geometry
Importance: High ✭✭✭
Author(s):
Subject: Geometry
Keywords: empty triangle
general position
ramsey theory
Recomm. for undergrads: no
Posted by: mdevos
on: October 8th, 2008

If $ X \subseteq {\mathbb R}^2 $ is a finite set of points which is 2-colored, an empty triangle is a set $ T \subseteq X $ with $ |T|=3 $ so that the convex hull of $ T $ is disjoint from $ X \setminus T $. We say that $ T $ is monochromatic if all points in $ T $ are the same color.

Conjecture   There exists a fixed constant $ c $ with the following property. If $ X \subseteq {\mathbb R}^2 $ is a set of $ n $ points in general position which is 2-colored, then it has $ \ge cn^2 $ monochromatic empty triangles.

It is known that any set of $ n $ points in the plane in general position contains $ \ge cn^{5/4} $ monochromatic empty triangles.

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