Monochromatic empty triangles

Definitions and mathematical statement

Let $P$ be a set of $n$ distinct points in the Euclidean plane in general position, meaning that no three lie on one line. Color each point either red or blue, with no assumption on the relative sizes of the color classes. For an unordered three-element subset $\{a,b,c\}\subseteq P$, its triangle is the closed convex hull of those points. Call it monochromatic when all three vertices have the same color, and empty when its interior contains no point of $P$. General position ensures that no other point lies on a triangle side, so this is equivalent to saying that the closed triangle meets $P$ in exactly its three vertices.

Write $E(P,c)$ for the number of such empty monochromatic triangles, counting vertex triples once and not distinguishing an orientation or a starting vertex. The conjecture is the existence of absolute constants $\gamma>0$ and $N\ge1$ such that \[ E(P,c)\ge\gamma n^2 \] for every $n\ge N$, every such planar point set, and every red--blue coloring $c$.

Both colors contribute to the count, and the emptiness test uses all points of both colors. In particular, a triangle whose interior contains points only of the opposite color is not empty for this problem. The asymptotic formulation allows exceptional small values of $n$; neither the constant nor the threshold may depend on the geometry or the coloring.

Short English statement

Must every sufficiently large two-colored planar point set in general position contain quadratically many empty triangles with one-colored vertices?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3081 (OPG-2435), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Monochromatic empty triangles. Original problem and discussion. http://www.openproblemgarden.org/op/monochromatic_empty_triangles.
• S3 J. Pach and G. Tóth, Monochromatic empty triangles in two-colored point sets, Discrete Applied Mathematics. https://doi.org/10.1016/j.dam.2011.08.026.