Saharon Shelah showed that if $\lambda$ is a regular cardinal with $2^\lambda=\lambda^+$, then adding $\lambda^{++}$-many Cohen reals yields a model in which there is no universal graph at $\lambda^+$ (see \cite{kojman}). Sy Friedman and Katherine Thompson \cite{friedman-thompson}, extended Shelah's result to the successor of a singular cardinal of countable cofinality, in partcular, assuming the existence of a strong cardinal, they have built a model in which $\aleph_\omega$ is strong limit and there are no universal graphs on $\aleph_{\omega+1}$.
In this talk I continue these works and sketch a proof of the following:
Theorem. Assuming the existence of a strong cardinal, one can build a model of ZFC in which the following holds: for any uncountable regular cardinal $\lambda,$ there is no universal graph on $\lambda$.