On the edit distance function of the random graph

Abstract Given a hereditary property of graphs $\mathcal{H}$ and $p\in [0,1]$ , the edit distance function $\textrm{ed}_{\mathcal{H}}(p)$ is asymptotically maximum proportion edge additions plus deletions applied to graph density p sufficient ensure that resulting satisfies . The directly related other well-studied quantities such as speed for -chromatic number random graph. Let be forbidding an Erdős–Rényi $F\sim \mathbb{G}(n_0,p_0)$ let $\varphi$ represent golden ratio. In this paper, we show if $p_0\in [1-1/\varphi,1/\varphi]$ then a.a.s. $n_0\to\infty$ \begin{align*} {\textrm{ed}}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{-\log(1-p_0)}, \frac{1-p}{-\log p_0} \right\}. \end{align*} Moreover, holds [1/3,2/3]$ any (0,1)$ A primary tool in proof categorization -core coloured regularity range $p\in[1-1/\varphi,1/\varphi]$ Such must have non-grey edges form vertex-disjoint cliques.