Combinatorial and computational problems about points in the plane

We study three problems in combinatorial geometry. The problems investigated are con ict-free colorings of point sets in the plane with few colors, polychromatic colorings of the vertices of rectangular partitions in the plane and in higher dimensions and polygonalizations of point sets with few re ex points. These problems are problems of discrete point sets, the proofs are of combinatorial avour with computational aspects and give e cient algorithms. First we give a historical introduction to the topics and place our results in this context. We also investigate the similarities between the proving methods of the three topics. In the rest of the Thesis we present the results in all detail including proofs.