Exactly $m$-coloured complete infinite subgraphs

TL;DR

This paper investigates the minimal number of colours needed to find an exactly m-coloured complete infinite subgraph within an edge-coloured infinite complete graph, establishing a tight lower bound related to the total number of colours used.

Contribution

It introduces a lower bound of at least rac{rac{1}{2}k}{rac{1}{2}} for the size of the set of such m, and proves this bound is tight for infinitely many k, advancing understanding of colour distributions in infinite graphs.

Findings

Set of such m has size at least rac{rac{1}{2}k}{rac{1}{2}}

Bound is tight for infinitely many values of k

Results extend to colourings with infinitely many colours

Abstract

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. The question of finding exactly $m$-coloured complete subgraphs was first considered by Erickson in 1994; in 1999, Stacey and Weidl partially settled a conjecture made by Erickson and raised some further questions. In this paper, we shall study, for a colouring of the edges of the complete graph on $\mathbb{N}$ with exactly $k$ colours, how small the set of natural numbers $m$ for which there exists an $m$-coloured complete infinite subgraph can be. We prove that this set must have size at least $\sqrt{2k}$; this bound is tight for infinitely many values of $k$. We also obtain a version of this result for colourings that use infinitely many colours.