Almost-spanning universality in random graphs

A graph $G$ is said to be $\mathcal{H}(n,\mathrm{\Delta })$-universal if it contains every graph on $n$ vertices with maximum degree at most $\mathrm{\Delta }$. It is known that for any $\epsilon >0$ and any natural number $\mathrm{\Delta }$ there exists $c

0$suchthattherandomgraph$G(n,p)$isasymptoticallyalmostsurely$\mathcal H((1-\varepsilon)n,\Delta)$-universalfor$p \geq c (\log n/n)^{1/\Delta}$\mathrm{.}Bypassingthisnaturalboundary,weshowthatfor$\Delta \geq 3$thesameconclusionholdswhen$p = \omega\left(n^{-\frac{1}{\Delta-1}}\log^5 n\right)$.