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Monochromatic empty triangles
If is a finite set of points which is 2-colored, an empty triangle is a set
with
so that the convex hull of
is disjoint from
. We say that
is monochromatic if all points in
are the same color.
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
![$ X \subseteq {\mathbb R}^2 $](/files/tex/71fbba54c5abad01c67f2c62eac8eb5eb7b71842.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ \ge cn^2 $](/files/tex/7cb1c285388b3509a0bafac8c671235c0a740e56.png)
It is known that any set of points in the plane in general position contains
monochromatic empty triangles.
Bibliography
* indicates original appearance(s) of problem.
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 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 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.
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.