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 $\alpha$.
Categories: math.CO