Abstract

We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal $\mu$, of cofinality $\omega$, such that every $\mu^{+}$ -chromatic graph X on $\mu^{+}$ has an edge colouring c of X into $\mu$ colours for which every vertex colouring g of X into at most $\mu$ 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 $\mu^{+}$ is replaced by other cardinals < $\mu$