The threshold conjecture for random graphs and critical subgraphs

Let $H$ be a graph with a color-critical edge $e$, meaning that $\chi(H-e)<\chi(H)$. For a graph $G$, let $t_H(G)$ and $b_H(G)$ be the maximum values of $|K|$ over, respectively, $H$-free and $(\chi(H)-1)$-partite subgraphs of $G$, and set

$$\mathcal F_H=\{t_H(G)=b_H(G)\}.$$

Say that $\mathcal G_H$ holds for $G$ if every edge of $G$ is color-critical in some copy of $H$ in $G$. For a graph property $\mathcal F$, write $p_c(\mathcal F)$ for its threshold in $G_{n,p}$. The threshold conjecture. For any $H$ with a color-critical edge,

$$p_c(\mathcal F_H)=O\bigl(p_c(\mathcal G_H)\bigr).$$

For cliques, this is the corresponding theorem; the threshold for $\mathcal G_H$ is known, while the conjectured comparison remains open in general. A stronger asymptotic equivalence $p_c(\mathcal F_H)\sim p_c(\mathcal G_H)$ is also suggested in the source.