Almost Empty Monochromatic Triangles in Planar Point Sets

For positive integers $c, s \geq 1$, let $M_3(c, s)$ be the least integer such that any set of at least $M_3(c, s)$ points in the plane, no three on a line and colored with $c$ colors, contains a monochromatic triangle with at most $s$ interior points. The case $s=0$, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that $M_3(1, 0)=3$, $M_3(2, 0)=9$ and $M_3(c, 0)=\infty$, for $c\geq 3$. In this paper we extend these results when $c \geq 2$ and $s \geq 1$. We prove that the least integer $\lambda_3(c)$ such that $M_3(c, \lambda_3(c))< \infty$ satisfies: $$\left\lfloor\frac{c-1}{2}\right\rfloor \leq\lambda_3(c)\leq c-2,$$ where $c \geq 2$. Moreover, the exact values of $M_3(c, s)$ are determined for small values of $c$ and $s$. We also conjecture that $\lambda_3(4)=1$, and verify it for sufficiently large Horton sets.