Given an edge colouring of a graph with a set of m colours, we say that the graph is m-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 ${\mathcal{F}}_{\mathrm{\Delta }}$ 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 ${\mathcal{F}}_{\mathrm{\Delta }}$ over all m-colourings Δ of ${\mathbb{N}}^{(2)}$. Indeed, we shall prove the following result. There exists an absolute constant $\alpha >0$ such that for any positive integer $m\ne \{\left(\begin{array}{c}n\\ 2\end{array}\right)+1,\left(\begin{array}{c}n\\ 2\end{array}\right)+2:n\ge 2\}$, $|{\mathcal{F}}_{\mathrm{\Delta }}|\ge (1+\alpha )\sqrt{2m}$, for any m-colouring Δ of ${\mathbb{N}}^{(2)}$. This proves a conjecture of Narayanan. We remark the result is tight up to the value of α.

给定的边缘与一组的一个图的着色米颜色，我们说的图表是米-着色如果每个的米被使用的颜色。对于m色差Δ${\mathbb{ñ}}^{（2）}$，完整的图形 $\mathbb{ñ}$，我们用 ${\mathcal{F}}_{\mathrm{\Delta }}$设置存在无限子集的所有值γ$X\subset \mathbb{ñ}$ 这样 $X^{（2）}$是γ色的。该集合的属性于1994年由Erickson首次研究。在这里，我们有兴趣估算${\mathcal{F}}_{\mathrm{\Delta }}$在所有m个色标Δ上${\mathbb{ñ}}^{（2）}$。确实，我们将证明以下结果。存在一个绝对常数$\alpha >0$ 这样对于任何正整数 $米\ne \{（\begin{array}{c}ñ\\ 2\end{array}）+\mathrm{1个}，（\begin{array}{c}ñ\\ 2\end{array}）+2：ñ\ge 2\}$， $|{\mathcal{F}}_{\mathrm{\Delta }}|\ge （\mathrm{1个}+\alpha ）\sqrt{2米}$，对于任何m色Δ${\mathbb{ñ}}^{（2）}$。这证明了纳拉亚南的猜想。我们注意到结果严格到α的值。