Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Ramsey Theory and Geometry of Closed Loops

Version 1 : Received: 11 February 2023 / Approved: 13 February 2023 / Online: 13 February 2023 (06:56:36 CET)

A peer-reviewed article of this Preprint also exists.

Shvalb, N.; Frenkel, M.; Shoval, S.; Bormashenko, E. A Note on the Geometry of Closed Loops. Mathematics 2023, 11, 1960. Shvalb, N.; Frenkel, M.; Shoval, S.; Bormashenko, E. A Note on the Geometry of Closed Loops. Mathematics 2023, 11, 1960.

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 yikx=αikx+βik i,k=1...6, αik≠0. We color the edges connecting the points for which αik>0 holds with red, and the edges for which αik<0 with green with red. At least one monochromic triangle should necessarily appear within the curve (according to the Ramsey number R3,3=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.

Keywords

Ramsey theory; closed contour; Jordan theorem; complete graph.

Subject

Computer Science and Mathematics, Mathematics

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