Monochromatic Empty Triangles in Two-Colored Point Sets

Abstract

A key problem in combinatorial geometry is the identification of properties of a subset of a point set which has properties like convexity, monochromaticity, and emptiness. This thesis actual works on this sides of combinatorial geometry: efficiently finding empty monochromatic triangles in two-colored point sets on the plane. While earlier research focused on counting empty triangles in point set without any color, our work takes an optimal algorithm to explicitly detect them. By using geometric insights and visibility-based techniques. We then address a relaxed but equally compelling variant: monochromatic triangles containing at most one point of the opposite color. For this problem, we derive a quadratic lower bound, revealing how even slight relaxations in geometric constraints dramatically influence lower bound of empty monochromatic triangly in two colored point set.