A stability theorem for maximal Kr+1-free graphs

For r≥2, we show that every maximal Kr+1-free graph G on n vertices with (1−1r)n22−o(nr+1r) edges contains a complete r-partite subgraph on (1−o(1))n vertices. We also show that this is best possible. This result answers a question of Tyomkyn and Uzzell.