Small universal families of graphs on ${\mathrm{\aleph }}_{\omega +1}$

We prove that it is consistent that ${\mathrm{\aleph }}_{\omega }$ is strong limit, $2^{{\mathrm{\aleph }}_{\omega }}$ is large and the universality number for graphs on ${\mathrm{\aleph }}_{\omega +1}$ is small. The proof uses Prikry forcing with interleaved collapsing.