Geometric Proof of a Ramsey-Type Result For Disjoint Empty Convex Polygons I

摘要

This is the first part of a two-part paper, where we give a geomet-ric proof of a Ramsey-type result for disjoint empty convex polygons.We prove that every set of 11 points in the plane, no three on a line,contains either an empty convex hexagon or an empty convex pentagonand a disjoint empty convex quadrilateral. This result was establishedby Aichholzer et al. [1] with the help of the order type data base [2, 3].In this two-part paper we give an intuitive geometric proof of this factbased on the analysis of a few distinct cases. In the first part, we provesome basic observations, following which we show that any set of 11points in the plane, no three on a line, with (a) at most six points inthe convex hull and exactly one point in the third convex layer, or (b)at least six points in the convex hull, contains an empty convex hexagonor an empty convex pentagon and a disjoint empty convex quadrilateral.