On the edit distance function of the random graph

Given a hereditary property of graphs $mathcal{H}$ and a $pin [0,1]$, the edit distance function ${rm ed}_{mathcal{H}}(p)$ is asymptotically the maximum proportion of edge-additions plus edge-deletions applied to a graph of edge density $p$ sufficient to ensure that the resulting graph satisfies $mathcal{H}$. The edit distance function is directly related to other well-studied quantities such as the speed function for $mathcal{H}$ and the $mathcal{H}$-chromatic number of a random graph. Let $mathcal{H}$ be the property of forbidding an ErdH{o}s-R{e}nyi random graph $Fsim mathbb{G}(n_0,p_0)$, and let $varphi$ represent the golden ratio. In this paper, we show that if $p_0in [1-1/varphi,1/varphi]$, then a.a.s. as $n_0toinfty$, begin{align*} {rm ed}_{mathcal{H}}(p) = (1+o(1)),frac{2log n_0}{n_0} cdotminleft{ frac{p}{-log(1-p_0)}, frac{1-p}{-log p_0} right}. end{align*} Moreover, this holds for $pin [1/3,2/3]$ for any $p_0in (0,1)$.