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Starting from a supercompact cardinal we build a model in which$2^{\aleph_{\omega_1}}=2^{\aleph_{\omega_1+1}}=\aleph_{\omega_1+3}$ but thereis a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on$\aleph_{\omega_1+1}$. The same technique will work for any uncountablecardinal in place of $\omega_1$.

Universal graphs at $\aleph_{\omega_1+1}$