Turán's theorem
In graph theory, Turán's theorem is a result on the number of edges in a Kr+1-free graph.
An n-vertex graph that does not contain any (r + 1)-vertex clique may be formed by partitioning the set of vertices into r parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. We call the resulting graph the Turán graph T(n, r). Turán's theorem states that the Turán graph has the largest number of edges among all Kr+1-free n-vertex graphs.
Turán graphs were first described and studied by Hungarian mathematician Pál Turán in 1941, though a special case of the theorem was stated earlier by Mantel in 1907.
An equivalent formulation is the following:
Let G be an n-vertex simple graph with no (r + 1)-clique and with the maximum number of edges.
Assume the claim is false. Construct a new n-vertex simple graph G′ that contains no (r + 1)-clique but has more edges than G, as follows:
Case 1:
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