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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 χ’ₖ(G). Let fₖ be defined by $fₖ\left(\Delta \right)=ma{x}_{G:\Delta \left(G\right)=\Delta }{\chi }^{\text{'}}ₖ\left(G\right)$. We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.