Almost Empty Monochromatic Polygons in Planar Point Sets

Abstract

For integers $k \ge 3$, $c \ge 2$, and $s \ge 0$, let $M^*_k(c, s)$ be the least integer such that any set of at least $M^*_k(c, s)$ points in the plane in general position, colored with c colors, contains a monochromatic k-gon (not necessarily convex) with at most s interior points. Denote by $\lambda _k^*(c)$ the least integer such that $M^*_k(c, \lambda _k^*(c)) < \infty$. It follows from results in [3, 5] that

$$ \left\lfloor \frac{c-1}{2} \right\rfloor \le \lambda _3^*(c) \le c-3. $$

In this paper, we extend this result to $k \ge 4$. Specifically, we show that, for $c\ge 3$,

$$ 2 \left\lfloor \frac{c-1}{2} \right\rfloor \le \lambda _4^*(c) \le 2 c-4. $$

Moreover, for $k \ge 5$ and $c \ge 2$, we show that

$$ (k-2) \left\lfloor \frac{c-1}{2} \right\rfloor \le \lambda _k^*(c) \le (k-2) c - (k-1). $$