A large number of $m$-coloured complete infinite subgraphs

TL;DR

This paper proves a lower bound on the size of the set of possible monochromatic subgraph sizes in any $m$-coloured infinite complete graph, confirming a conjecture and extending understanding of colourings in infinite combinatorics.

Contribution

It establishes a nearly optimal lower bound on the minimum size of the set of achievable monochromatic subgraph sizes in any $m$-colouring of an infinite complete graph, confirming a conjecture by Narayanan.

Findings

Proves a lower bound of approximately rom the abstractor rom the abstractor all but finitely many $m$.

Confirms a conjecture of Narayanan regarding the size of rom the abstractor all $m$-colourings.

Shows the bound is tight up to a constant factor.

Abstract

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is $m$-\textit{coloured} if each of the $m$ colours is used. For an $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, the complete graph on $\mathbb{N}$, we denote by $\mathcal{F}_{\Delta}$ the set all values $\gamma$ for which there exists an infinite subset $X\subset \mathbb{N}$ such that $X^{(2)}$ is $\gamma$-coloured. Properties of this set were first studied by Erickson in $1994$. Here, we are interested in estimating the minimum size of $\mathcal{F}_{\Delta}$ over all $m$-colourings $\Delta$ of $\mathbb{N}^{(2)}$. Indeed, we shall prove the following result. There exists an absolute constant $\alpha > 0$ such that for any positive integer $m \neq \left\{ {n \choose 2}+1, {n \choose 2}+2: n\geq 2\right\}$, $|\mathcal{F}_{\Delta}| \geq (1+\alpha)\sqrt{2m}$, for any $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, thus proving a conjecture of Narayanan. This result is tight up to the order of the constant $\alpha$.