We propose the following problem. 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 colouring. The minimum number of colours required for such a colouring is the k- intersection chromatic index and is denoted χ 0 k . Let f be defined by fk(∆) = max G:∆(G)=∆ {χ 0 k (G)} We show that fk(∆) = Θ( ∆2 k ). We also discuss some open problems.