Turán's theorem

Turán's theorem is a fundamental result in extremal graph theory that provides a bound on the number of edges in a graph that avoids complete subgraphs (cliques) of a given size. Specifically, it deals with the maximum number of edges that can be present in a graph with \( n \) vertices that does not contain a complete subgraph \( K_{r+1} \) (a complete graph on \( r+1 \) vertices).