On a Problem of 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).