Almost Empty Monochromatic Quadrilaterals in Planar Point Sets

For positive integers c, s 1, r 3, 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)=W (r) (c, s). In this paper, we extend these results to the case r = 4. We prove that W4(2, 1) = 11, W4(3, 2) 120, and the least integer mu 4(c) such that W4(c, mu 4(c)) < a is bounded by ,where c >= 2.