The threshold conjecture of Turán properties for random graphs is true

You need to know: Graph, notation $K_n$ for complete graph on n vertices, notations $v(H)$ and $e(H)$ for the number of vertices and edges in graph H, respectively.

Background: For a graph $F$ with $e(F)\geq 1$ and $v(F)\geq 3$ , let $m(F)$ be the maximum value of $\frac{e(H)-1}{v(H)-2}$ over all subgraphs H of F with $v(H)\geq 3$ . We say that a graph is F-free if it does not contain a copy of a graph F as a subgraph. For graph G, we write $\text{ex}(G, F)$ for the maximal number of edges that an F-free subgraph of G may have. Number $\pi(F)=\lim\limits_{n\to\infty}\frac{ex(K_n,F)}{e(K_n)}$ is known as Turán density of graph F. The Erdős-Renyi graph $G(n,p)$ is a random graph with n vertices, such that every possible edge is included independently with probability $p=p(n)$ .

The Theorem: On 16th October 2009, Mathias Schacht submitted to the Annals of Mathematics a paper in which he proved that for every graph $F$ with $e(F)\geq 1$ and $v(F)\geq 3$ and every $\epsilon>0$ , there exists $C=C(\epsilon)>0$ such that for every sequence of probabilities $q_n \geq Cn^{-1/m(F)}$ , we have $\lim\limits_{n\to\infty} P\left[\text{ex}(G(n,q_n),F) \leq \left(\pi(F)+\epsilon\right)e(G(n,q_n))\right]=1$ .

Short context: Function $\text{ex}(G, F)$ was first studied by Mantel, Erdős, and Turán. In particular, Turán determined in 1941 $\text{ex}(K_n, K_k)$ for all integers n and k. For general graph G, it is known that $\text{ex}(G, F) \geq \pi(F)e(G)$ . The Theorem proves that if G is a random graph, then the opposite inequality $\text{ex}(G, F) \leq (\pi(F)+\epsilon)e(G)$ holds with high probability. It confirms a conjecture of Kohayakawa, Łuczak, and Roedl made in 1997.