Turán's theorem

In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that does not have a given subgraph.

An example of an ${\displaystyle n}$-vertex graph that does not contain any ${\displaystyle (r+1)}$-vertex clique ${\displaystyle K_{r+1}}$ may be formed by partitioning the set of ${\displaystyle n}$ vertices into ${\displaystyle r}$ parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the Turán graph ${\displaystyle T(n,r)}$. Turán's theorem states that the Turán graph has the largest number of edges among all K_r+1-free n-vertex graphs.

Turán's theorem, and the Turán graphs giving its extreme case, were first described and studied by Hungarian mathematician Pál Turán in 1941. The special case of the theorem for triangle-free graphs is known as Mantel's theorem; it was stated in 1907 by Willem Mantel, a Dutch mathematician.