Universal graphs at ${\mathrm{\aleph }}_{{\omega }_1+1}$

Starting from a supercompact cardinal we build a model in which $2^{{\mathrm{\aleph }}_{{\omega }_1}}=2^{{\mathrm{\aleph }}_{{\omega }_1+1}}={\mathrm{\aleph }}_{{\omega }_1+3}$ but there is a jointly universal family of size ${\mathrm{\aleph }}_{{\omega }_1+2}$ of graphs on ${\mathrm{\aleph }}_{{\omega }_1+1}$. The same technique will work for any uncountable cardinal in place of ${\omega }_1$.