Wild edge colourings of graphs

We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal μ , of cofinality ω , such that every μ + -chromatic graph X on μ + has an edge colouring c of X into μ colours for which every vertex colouring g of X into at most μ many colours has a g -colour class on which c takes every value. The paper also contains some generalisations of the above statement in which μ + is replaced by other cardinals > μ .