Almost Empty Monochromatic Polygons in Planar Point Sets

For integers $r \ge 3$, $c \ge 2$, and $s \ge 0$, let $\mathsf{M}_r^{\star}(c,s)$ denote the smallest integer such that any set of at least $\mathsf{M}_r^{\star}(c,s)$ points in the plane in general position (no three on a line), colored with $c$ colors, contains a monochromatic $r$-gon (not necessa