This is talk 4 out of 4. Last time we introduced a set-theoretic principle S(kappa,theta), which implies a particular anti-Ramsey theoretic result on all groups of cardinality kappa. In this talk we will prove that S(omega_1,omega_1) holds, and also that S(kappa,omega) holds for kappa equal to the cofinality of the continuum (this last result has a nice implication for the additive group of real numbers). Speaker(s): David Fernandez Breton (University of Michigan)