Korándi Dániel: Exact stability for Turán's theorem

Turán's theorem says that an extremal K_{r+1}-free graph is r-partite. The Stability Theorem of Erdős and Simonovits shows that if a K_{r+1}-free graph with n vertices has close to the maximal t_r(n) edges, then it is close to being r-partite.
In this talk we determine exactly the K_{r+1}-free graphs that are farthest from being r-partite among the ones with at least t_r(n) - \eps n^2 edges, for every small enough \eps. This extends work by Erdős, Győri and Simonovits, and proves a conjecture of Balogh, Clemen, Lavrov, Lidický and Pfender.
Joint work with Alexander Roberts and Alex Scott.