In this paper, we consider the problem of counting almost empty monochromatic triangles in colored planar point sets, that is, triangles whose vertices are all assigned the same color and that contain only a few interior points. Specifically, we show that any $c$-coloring of a set of $n$ points in the plane in general position (that is, no three on a line) contains $Ω(n^2)$ monochromatic triangles with at most $c-1$ interior points and $Ω(n^{\frac{4}{3}})$ monochromatic triangles with at most $c-2$ interior points, for any fixed $c \geq 2$. The latter, in particular, generalizes the result of Pach and Tóth (2013) on the number of monochromatic empty triangles in 2-colored point sets, to the setting of multiple colors and monochromatic triangles with a few interior points. We also derive the limiting value of the expected number of triangles with $s$ interior points in random point sets, for any integer $s \geq 0$. As a result, we obtain the expected number of monochromatic triangles with at most $s$ interior points in random colorings of random point sets.

翻译：本文研究了着色平面点集中几乎空单色三角形的计数问题，即所有顶点被赋予相同颜色且仅包含少量内部点的三角形。具体而言，我们证明：对于任意固定 $c \geq 2$，处于一般位置（即任意三点不共线）的 $n$ 个平面点集的 $c$ 着色方案中，必然存在 $Ω(n^2)$ 个至多包含 $c-1$ 个内部点的单色三角形，以及 $Ω(n^{\frac{4}{3}})$ 个至多包含 $c-2$ 个内部点的单色三角形。特别地，后一结论将 Pach 和 Tóth（2013）关于双色点集中空单色三角形数量的结果，推广至多色情形及包含少量内部点的单色三角形场景。此外，我们推导了随机点集中包含 $s$ 个内部点的三角形期望数量的极限值（$s$ 为任意非负整数）。基于此，我们进一步得到了随机点集的随机着色方案中，至多包含 $s$ 个内部点的单色三角形的期望数量。