Kahn–Kalai threshold conjecture for graph appearances

Let $G$ be a graph, and write $v(G)$ and $e(G)$ for its numbers of vertices and edges. For a subgraph $F\subseteq G$, define $p_E(G)$ to be the smallest $p$ such that

$$\frac{v(F)!}{|\operatorname{Aut}(F)|}p^{e(F)}\geq 1$$

for every $F\subseteq G$. A threshold function for a sequence of graphs $\{G_n\}$ is a function ${\rm th}(G_n)$ such that, for the random graph $G(n,p(n))$, the appearance probability tends to one when $p(n)/{\rm th}(G_n)\to\infty$ and tends to zero when $p(n)/{\rm th}(G_n)\to 0$. Kahn–Kalai threshold conjecture. There is a universal constant $K$ such that, for every sequence $\{G_n\}$, a threshold function for the appearance of $\{G_n\}$ in $G(n,p)$ is at most

$$p_E(G_n)\log |v(G_n)|.$$

This conjecture proposes that the first-moment obstruction determines thresholds up to a logarithmic factor, uniformly over all graph sequences. It was proposed by Kahn and Kalai; the supplied text gives no resolution status.