Ramsey Theory and Geometry of Closed Loops

TL;DR

This paper applies Ramsey theory to analyze geometric properties of closed contours, demonstrating the inevitability of monochromatic triangles and exploring relations within regions, with implications for dynamical billiards and differential geometry.

Contribution

It introduces novel applications of Ramsey theory to closed curves and their geometric properties, linking combinatorics with differential geometry and dynamical systems.

Findings

Monochromatic triangles necessarily appear in point sets on closed contours.

Ramsey constructions relate to differential geometry of closed contours.

Results have implications for analyzing dynamical billiards.

Abstract

We apply the Ramsey theory to the analysis of geometrical properties of closed contours. Consider a set of six points placed on a closed contour. The straight lines connecting these points are y_ik (x)={\alpha}_ik x+\b{eta}_ik (i,k=1...6), {\alpha}_ik is not equal to 0. We color the edges connecting the points for which {\alpha}_ik>0 holds with red, and the edges for which {\alpha}_ik<0 with green with red. At least one monochromic triangle should necessarily appear within the curve (according to the Ramsey number R(3,3)=6). This result is immediately applicable for the analysis of dynamical billiards. The second theorem emerges from the combination of the Jordan curve and Ramsey theorem. The closed curve is considered. We connect the points located within the same region with green links and the points placed within the different regions with red links. In this case, transitivity/intransitivity of the relations between the points should be considered. Ramsey constructions arising from the differential geometry of closed contours are discussed.