A stability theorem for multi-partite graphs

The Erdős-Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős-Simonovits type stability theorem in multi-partite graphs. Different from the Erdős-Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an $H$-free graph $G$ is close to the extremal graphs for $H$, then $G$ has a well-defined structure but may be far away to the extremal graphs for $H$. As an application, we solve a conjecture posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique

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