Almost Empty Monochromatic Quadrilaterals in Planar Point Sets

For positive integers $c,s\geq1$, $r\geq3$, let $W_r(c,s)$ be the least integer such that if a set of at least $W_r(c,s)$ points in the plane, no three of which are collinear, is colored with $c$ colors, then this set contains a monochromatic $r$-gon with at most $s$ interior points. As is known, if $r=3$, then $W_r(c,s)$=$M_r(c,s)$. In this paper, we extend these results to the case $r=4$. We prove that $W_4(2,1)=11$, $W_4(3,2)\leq120$, and the least integer $\mu_4(c)$ such that $W_4(c,\mu_4(c))<\infty$ is bounded by
$$ \big\lfloor\frac{c-1}{2}\big\rfloor\cdot2\leq\mu_4(c)\leq2c-3, \qquad\text{where}\quad c\geq2. $$