Universal graphs at $\aleph_{ω_1+1}$

Starting from a supercompact cardinal we build a model in which $2^{\aleph_{ω_1}}=2^{\aleph_{ω_1+1}}=\aleph_{ω_1+3}$ but there is a jointly universal family of size $\aleph_{ω_1+2}$ of graphs on $\aleph_{ω_1+1}$. The same technique will work for any uncountable cardinal in place of $ω_1$.

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