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.
"Wild edge colourings of graphs." J. Symbolic Logic 69 (1) 255 - 264, March 2004. https://doi.org/10.2178/jsl/1080938840