The Calkin algebra is ${\mathrm{\aleph }}_1$-universal

We discuss the existence of (injectively) universal C*-algebras and prove that all C*-algebras of density character ${\mathrm{\aleph }}_1$ embed into the Calkin algebra, $Q(H)$. Together with other results, this shows that each of the following assertions is relatively consistent with ZFC: (i) $Q(H)$ is a $2^{{\mathrm{\aleph }}_0}$-universal C*-algebra. (ii) There exists a $2^{{\mathrm{\aleph }}_0}$-universal C*-algebra, but $Q(H)$ is not $2^{{\mathrm{\aleph }}_0}$-universal. (iii) A $2^{{\mathrm{\aleph }}_0}$-universal C*-algebra does not exist. We also prove that it is relatively consistent with ZFC that (iv) there is no ${\mathrm{\aleph }}_1$-universal nuclear C*-algebra, and that (v) there is no ${\mathrm{\aleph }}_1$-universal simple nuclear C*-algebra.