Monochromatic triangles in three-coloured graphs

James Cummings,Daniel Králʼ,Florian Pfender,Konrad Sperfeld,Andrew Treglown,Michael Young

doi:10.1016/j.jctb.2013.05.002

Abstract

In 1959, Goodman [9] determined the minimum number of monochromatic triangles in a complete graph whose edge set is 2-coloured. Goodman (1985) [10] also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, for n sufficiently large, we determine the minimum number of monochromatic triangles in a 3-coloured copy of Kn. Moreover, we characterise those 3-coloured copies of Kn that contain the minimum number of monochromatic triangles.

Highlights

The Ramsey number Rk(G) of a graph G is the minimum n ∈ N such that every k-colouring of Kn contains a monochromatic copy of G
The Ramsey multiplicity Mk(G, n) of G is the minimum number of monochromatic copies of G over all k-colourings of Kn. (Here, we are counting unlabelled copies of G in the sense that we count the number of distinct monochromatic subgraphs of Kn that are isomorphic to G.) In the case when k = 2 we write M (G, n)
Given r ∈ N, we denote the complete graph on r vertices by Kr and define R(r, r) := R2(Kr)

Summary

Introduction

The Ramsey number Rk(G) of a graph G is the minimum n ∈ N such that every k-colouring of Kn contains a monochromatic copy of G. The Ramsey multiplicity Mk(G, n) of G is the minimum number of monochromatic copies of G over all k-colourings of Kn. A graph G is k-common if Mk(G, n) asymptotically equals, as n tends to infinity, the expected number of monochromatic copies of G in a random k-colouring of Kn. Erdos [6] conjectured that. The focus of this paper is to give the exact value of M3(K3, n) for sufficiently large n, thereby yielding a 3-coloured analogue of Goodman’s theorem. We characterise those 3-coloured copies of Kn that contain exactly M3(K3, n) monochromatic triangles. Is Mk(K3, n) equal to the number of monochromatic triangles in Gex(n, k)?

Notation

Graph densities

Flag algebras

Proof of Theorem 2