Version 1
: Received: 8 June 2023 / Approved: 9 June 2023 / Online: 9 June 2023 (09:48:51 CEST)
How to cite:
Frenkel, M.; Shoval, S.; Bormashenko, E. Ramsey Theory and Transformations of Coordinate Systems. Preprints2023, 2023060685. https://doi.org/10.20944/preprints202306.0685.v1
Frenkel, M.; Shoval, S.; Bormashenko, E. Ramsey Theory and Transformations of Coordinate Systems. Preprints 2023, 2023060685. https://doi.org/10.20944/preprints202306.0685.v1
Frenkel, M.; Shoval, S.; Bormashenko, E. Ramsey Theory and Transformations of Coordinate Systems. Preprints2023, 2023060685. https://doi.org/10.20944/preprints202306.0685.v1
APA Style
Frenkel, M., Shoval, S., & Bormashenko, E. (2023). Ramsey Theory and Transformations of Coordinate Systems. Preprints. https://doi.org/10.20944/preprints202306.0685.v1
Chicago/Turabian Style
Frenkel, M., Shraga Shoval and Edward Bormashenko. 2023 "Ramsey Theory and Transformations of Coordinate Systems" Preprints. https://doi.org/10.20944/preprints202306.0685.v1
Abstract
Application of the Ramsey Theory to the set of the points placed in the plane is discussed. Consider a set of N points located in the same plane. The straight lines connecting these points are yikx=αikx+βik i,k=1...N. Listed below values of slopes αik are possible: αik>0; αik=0; αik is not defined; αik<0. Following coloring procedure is introduced: we connect the pairs of points for which αik>0, αik=0 or αik is not defined, with the red links, and the pairs of points for which αik<0 place with green links. The suggested coloring procedure enables building of the complete bi-colored graph for any set containing N points located in the same plane. We apply the Ramsey theorem to the complete graph emerging from the suggested coloring. For the set containing N=6 points at least one monochromatic triangle will necessarily appear in the graph. The values of the slopes αik depend on the chosen coordinate system. The rotation of coordinate axes changes the coloring of the graph; however, at least one monochromatic triangle will be present for N=6. The introduced coloring procedure diminishes the order of the symmetry group of the regular hexagon, irrespectively to the orientation of the coordinate axes. We hypothesized that this will be true for arbitrary regular n-polygon, independently on the orientation of coordinate axes. The inverse bi-color Ramsey graphs arise, when we replace red links appearing in the source graph with red ones, and vice versa. The total number of triangles in the “direct” and “inverse” Ramsey graphs is the same. We considered the particular case of the set of points for which αik>0 or αik<0 is true in the given coordinates. In this particular case, the rotation of the Cartesian coordinate axes to the angle θ=π2 yields the inverse complete graph. Generalization of this coloring for an arbitrary number and arbitrary location of the source points is introduced.
Computer Science and Mathematics, Discrete Mathematics and Combinatorics
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.
