We propose the following problem. For some k ≥ 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sufficient to guarantee such a colouring is the k-intersection chromatic index and is denoted Xik(G). Let fk be defined by fk ($\delta$) = max/ G:$\delta$(G)=$\delta$ {Xik(G)}. We show that fk($\delta$) = $\theta$($\delta$ 2/k ). We also discuss some open problems.