Monochromatic Empty Triangles

Definitions and mathematical statement

For each integer $n\geq 1$, let $\mathcal C_n$ be the collection of pairs $(R,B)$ of disjoint finite subsets of ${\mathbb{R}}^2$ such that $|R|=|B|=n$ and no three points of $P=R\cup B$ are collinear. Points of $R$ are red and points of $B$ are blue; here general position is interpreted as the stated noncollinearity condition.

A triangle determined by $P$ is the convex hull $\operatorname{conv}\{x,y,z\}$ of an unordered three-element subset $\{x,y,z\}\subseteq P$. It is monochromatic if all three vertices belong to $R$, or all three belong to $B$. It is empty if \[ \operatorname{int}\left(\operatorname{conv}\{x,y,z\}\right)\cap P=\varnothing. \] Thus emptiness excludes interior points of either color. The interior is the ordinary planar interior; the general-position assumption also excludes additional points on the edges.

Let $T(R,B)$ denote the number of monochromatic empty triangles, counting each unordered vertex triple once, and define \[ f(n)=\min_{(R,B)\in\mathcal C_n}T(R,B). \] Equivalently, $f(n)$ is the largest nonnegative integer such that every configuration in $\mathcal C_n$ contains at least $f(n)$ monochromatic empty triangles.

Problem. Prove or disprove that \[ \lim_{n\to\infty}\frac{f(n)}{n^2}=0. \] Explicitly, the proposed assertion is \[ \forall\varepsilon>0\ \exists N\geq1\ \forall n\in{\mathbb{Z}}_{\geq1},\quad n\geq N\ \Longrightarrow\ \exists(R,B)\in\mathcal C_n:\ T(R,B)<\varepsilon n^2. \]

Source qualification. This formulation follows the supplied reconstructed statement. The supplied original excerpt begins mid-sentence, uses both $m(n)$ and $f(n)$ without a complete definition, and continues into unrelated problems. It therefore does not by itself establish the complete formulation above. The registry reports that the reconstruction was checked against the official report; that reported check has not been independently verified here.

Short English statement

Among all general-position configurations with exactly $n$ red and $n$ blue points, let $f(n)$ be the smallest number of monochromatic triangles whose interiors contain no point of either color. Determine whether $f(n)/n^2$ tends to zero. The supplied registry labels the problem open; this draft preserves that label without independently verifying its current status.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30005820$, title Monochromatic Empty Triangles, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2024; no proposer is supplied for this problem. The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied curation claims, not independent verification by this draft.
• Discrete Geometry (2024), Oberwolfach Reports, DOI 10.4171/owr/2024/3, pp. 137--202. Supplied report provenance: https://doi.org/10.4171/owr/2024/3. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Pach and Toth, Monochromatic empty triangles in two-colored point sets. Supplied literature reference: https://doi.org/10.1016/j.dam.2011.08.026.
• Oberwolfach Report 3/2024. Supplied literature-source link: https://ems.press/content/serial-article-files/48651.
• Supplied literature assessment dated 2026-08-22: the problem is labeled open; Pach and Toth are credited with a lower bound of order $n^{4/3}$, and the best general upper bound is described as quadratic. The supplied assessment concludes that the requested $o(n^2)$ upper bound remains unresolved. It also calls the $n^{4/3}$ lower bound "quadratic-scale," an unclear description retained here as a source qualification. This assessment is supplied dated curation derived from the OWR record, not independently verified; its status and cited claims require independent checking.