Exact stability for Turán’s Theorem

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability of Erd\H{o}s and Simonovits shows if a with $n$ vertices has close to the maximal $t_r(n)$ edges, then it being In this paper we determine exactly graphs at least $m$ edges are farthest from $r$-partite, for any $m\ge t_r(n) - \delta_r n^2$. This extends work by Erd\H{o}s, Gy\H{o}ri Simonovits, proves conjecture Balogh, Clemen, Lavrov, Lidick\'y Pfender.