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A stability theorem for multi-partite graphs
- Part of
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 11 August 2025, pp. 1-27
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- Article
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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 from the extremal graphs for 
$H$. As applications, we strengthen a theorem of Bollobás, Erdős, and Straus and solve a conjecture in a stronger form 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.
