We propose the following problem. For some \(k\geq 1\), 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 \(\chi_k'(G)\). Let \(f_k\) be defined by \[ f_k(\Delta)= \max_{G:\Delta(G)=\Delta} \{\chi_k'(G)\}. \] We show that \(f_k(\Delta)= \Theta(\frac{\Delta^2}{k})\). We also discuss some open problems.