empty triangle

Monochromatic empty triangles โ˜…โ˜…โ˜…


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}

Keywords: empty triangle; general position; ramsey theory

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