# Monochromatic empty triangles

 Importance: High ✭✭✭
 Author(s):
 Subject: Geometry
 Keywords: empty triangle general position ramsey theory
 Posted by: mdevos on: October 8th, 2008

If $X \subseteq {\mathbb R}^2$ is a finite set of points which is 2-colored, an \emph{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 \emph{monochromatic} if all points in $T$ are the same color.

\begin{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. \end{conjecture}

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

## Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.

### This has a trivial

This has a trivial counterexample for c > 0.

Consider X = {(0,0), (0,1), (1,0)}, colored {red, blue, blue} respectively. There is only one empty triangle in X, and it is not monochromatic. So it has 0 monochromatic empty triangles, and 0 is not > c*(3^2) for c > 0.

### Yes indeed. However in this

Yes indeed. However in this types of problems it is generally implied that the statement is for a sufficently large n.

### Original source and one improvement.

The conjecture appeared first in "Oswin Aichholzer, Ruy Fabila-Monroy, David Flores-Peñaloza, Thomas Hackl, Clemens Huemer, and Jorge Urrutia. Empty monochromatic triangles. In Proceedings of the 20th Canadian Conference on Computational Geometry (CCCG2008), pages 75-78, 2008."

In this paper the authors show that any set of n points in general position has $cn^{5/4}$ empty monochromatic triangles. You can get this paper from http://cccg.ca/proceedings/2008/paper18.pdf

There is one improvement showing that any set of n points in general position has $cn^{4/3}$ empty monochromatic triangles in: "J. Pach, G. Toth. Monochromatic empty triangles in two-colored point sets. In: Geometry, Games, Graphs and Education: the Joe Malkevitch Festschrift (S. Garfunkel, R. Nath, eds.), COMAP, Bedford, MA, 2008, 195--198." Get it from: http://www.math.nyu.edu/~pach/publications/emptytriangle102408.pdf

### The lower bound has been improved

The lower bound has been improved to cn4/3.

J. Pach and G. Toth: Monochromatic empty triangles in two-colored point sets, in: Geometry, Games, Graphs and Education: the Joe Malkevitch Festschrift (S. Garfunkel, R. Nath, eds.), COMAP, Bedford, MA, 2008, 195--198.