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