  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 $\alpha$.
