Alex Riasanovsky: Partitioning random graphs with Turán graph

In 1987, Bollobás showed that the chromatic number of a random graph $G(n,p)$ is asymptotically almost surely equal to $(1+o(1))\dfrac{n}{\log_b(n)}$ where $b = 1/(1-p)$.  Essential to the proof was a sharp concentration of the clique number $\omega$ number of $G(n,p)$.  Motivated by a problem of computing the edit distance function of a random graph, we prove a generalization where "clique" is replaced by induced copies of (the complement of) the $t$-partite Turán graph.  This result matches a special case of a weaker concentration result due to Bollobás and Thomason in 2000 where "clique" is instead replaced by any induced subgraph from a hereditary property.