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

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is $m$-$\mathit{\text{coloured}}$ if each of the $m$ colours is used. For an $m$-colouring $\mathrm{\Delta }$ of ${\mathbb{N}}^{(2)}$, the complete graph on $\mathbb{N}$, we denote by ${\mathcal{F}}_{\mathrm{\Delta }}$ 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 ${\mathcal{F}}_{\mathrm{\Delta }}$ over all $m$-colourings $\mathrm{\Delta }$ 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}$_{\Delta}| \geq (1+\alpha)\sqrt{2m}$,forany$m$-colouring$\Delta$of$$\mathbb{N}$^{(2)}$, thus proving a conjecture of Narayanan. This result is tight up to the order of the constant $\alpha$.