  We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph
$G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that
almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor
of $G$, where $\lambda=0.63817\dots$ is an explicitly defined constant. This
generalises analogous results for complete graphs by Thomason (2001) and for
general dense graphs by Myers and Thomason (2005). It also shows that an upper
bound for sparse graphs by Reed and Wood (2016) is best possible up to a
constant factor.
