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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
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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