Monochromatic triangles in three-coloured graphs

TL;DR

This paper determines the minimum number of monochromatic triangles in large complete graphs with three colours and identifies the colourings that achieve this minimum, extending Goodman’s two-colour results.

Contribution

It provides the first exact results for the minimum monochromatic triangles in three-coloured complete graphs and characterizes the extremal colourings.

Findings

Minimum monochromatic triangles in three-coloured complete graphs identified

Extremal colourings achieving the minimum characterized

Results extend classical two-colour case to three colours

Abstract

In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph.