  A graph $G$ is said to be $\mathcal H(n,\Delta)$-universal if it contains
every graph on $n$ vertices with maximum degree at most $\Delta$. It is known
that for any $\varepsilon > 0$ and any natural number $\Delta$ there exists $c
> 0$ such that the random graph $G(n,p)$ is asymptotically almost surely
$\mathcal H((1-\varepsilon)n,\Delta)$-universal for $p \geq c (\log
n/n)^{1/\Delta}$. Bypassing this natural boundary, we show that for $\Delta
\geq 3$ the same conclusion holds when $p =
\omega\left(n^{-\frac{1}{\Delta-1}}\log^5 n\right)$.
