On a Problem of Ramsey Theory

TL;DR

This paper explores the minimum number of monochromatic triangles in a 3-coloring of the complete graph on 17 vertices, building on classical results in Ramsey theory and examining colorings of K17.

Contribution

It investigates the lower bounds on monochromatic triangles in 3-colorings of K17, extending the understanding of Ramsey numbers and graph colorings.

Findings

Proves the existence of at least one monochromatic triangle in any 3-coloring of K17.

Analyzes the minimum number of such triangles across all possible colorings.

Provides insights into the structure of colorings related to Ramsey theory.

Abstract

In 1955, Greenwood and Gleason showed that the Ramsey number R(3, 3, 3) = 17 by constructing an edge-chromatic graph on 16 vertices in three colors with no triangles. Their technique employed finite fields. This same result was obtained later by using another technique. In this article, we examine the complete graph on 17 vertices, K17, which can be represented as a regular polygon of 17 sides with all its diagonals. We color each edge of K17 with one of the three colors, blue, red or yellow. The graph thus obtained is called complete trichromatic graph K17^(3) (the superscript determines the number of colors). A triangle contained in graph K17^(3) with edges colored with one and only one color is called monochromatic. It has been shown that for any coloring of the K17^(3) edges, K17^(3) contains at least one monochromatic triangle. This article examines the problem of determining the minimum number of monochromatic triangles with the same color contained in K17^(3).