A Note on the Geometry of Closed Loops

Autor: Nir Shvalb, Mark Frenkel, Shraga Shoval, Edward Bormashenko
Jazyk: angličtina
Rok vydání: 2023
Předmět:
Zdroj: Mathematics, Vol 11, Iss 8, p 1960 (2023)
Druh dokumentu: article
ISSN: 2227-7390
DOI: 10.3390/math11081960
Popis: In this paper, we utilize the Ramsey theory to investigate the geometrical characteristics of closed contours. We begin by examining a set of six points arranged on a closed contour and connected as a complete graph. We assign the downward-pointing edges a red color, while coloring the remaining edges green. Our analysis establishes that the curve must contain at least one monochromatic triangle. This finding has practical applications in the study of dynamical billiards. Our second result is derived from the Jordan curve theorem and the Ramsey theorem. Finally, we discuss Ramsey constructions arising from differential geometry. Applications of the Ramsey theory are discussed.
Databáze: Directory of Open Access Journals
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