Robustness of the Sauer-Spencer Theorem

We prove a robust version of a graph embedding theorem of Sauer and Spencer. To state this sparser analogue, we define $G(p)$ to be a random subgraph of $G$ obtained by retaining each edge of $G$ independently with probability $p \in [0,1]$, and let $m_1(H)$ be the maximum $1$-density of a graph $H$. We show that for any constant $\Delta$ and $\gamma > 0$, if $G$ is an $n$-vertex host graph with minimum degree $\delta(G) \geq (1 - 1/2\Delta + \gamma)n$ and $H$ is an $n$-vertex graph with maximum degree $\Delta(H) \leq \Delta$, then for $p \geq Cn^{-1/m_1(H)}\log n$, the random subgraph $G(p)$ contains a copy of $H$ with high probability. Our value for $p$ is optimal up to a log-factor. In fact, we prove this result for a more general minimum degree condition on $G$, by introducing an \emph{extension threshold} $\delta_{\rm e}(\Delta)$, such that the above result holds for graphs $G$ with ${\delta(G) \geq (\delta_{\rm e}(\Delta) + \gamma)n}$. We show that $\delta_{\rm e}(\Delta) \leq (2\Delta-1)/2\Delta$, and further conjecture that $\delta_{\rm e}(\Delta)$ equals $\Delta/(\Delta+1)$, which matches the minimum degree condition on $G$ in the Bollob\'as-Eldridge-Catlin Conjecture. A main tool in our proof is a vertex-spread version of the blow-up lemma of Allen, B\"{o}ttcher, H\`{a}n, Kohayakawa, and Person, which we believe to be of independent interest.