On Almost Empty Monochromatic Triangles and Convex Quadrilaterals in Colored Point Sets

Let S be a set of N points on the plane in general position, colored arbitrarily with c colors (\(c\in {\mathbb {N}}\)). A subset of points is said to be monochromatic if all its points have the same color. In this paper we show that if N is sufficiently large, then S contains a monochromatic triangle with vertices in S and containing at most \(c-3\) points in its interior \((c \ge 4)\). The particular case \(c=4\) solves the conjecture of Basu, Bhattacharya, and Das in [6]. For the case of two colors, it is still unknown whether every sufficiently large bichromatic point set contains an empty monochromatic convex quadrilateral [11]. In this direction we show that if S is sufficiently large, then it always contains a convex monochromatic quadrilateral with at most \(2c-3\) points in its interior \((c \ge 2)\). We also show a general result on the number of empty convex quadrilaterals with disjoint interiors. Let P be a set of n points in general position. It is straightforward to see that if the elements of P are the vertices of a convex polygon, \(n \ge 4\), P always has \(\left\lceil \frac{n-3}{2} \right\rceil \) empty convex quadrilaterals with disjoint interiors. In this paper we prove that any point set P in general position has at least \(\left\lfloor \frac{n-3}{2}\right\rfloor \) interior disjoint empty convex quadrilaterals. We also give direct consequences of this theorem related to simultaneously flippable edges in triangulations and convex decompositions of point sets. We show that for any point set P there is always a triangulation that has \(\left\lfloor \frac{n-3}{2}\right\rfloor \) simultaneously flippable edges. We also show that there is always a convex decomposition of P consisting of triangles and convex quadrilaterals with at most \({3n-2h\over 2}\) elements, where h is the number of points in the convex hull of P.