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.
Let G be any subgraph of Kn such that G is Kr+1 -free. Then the number of edges in G is at most
An equivalent formulation is the following:
Among the n-vertex simple graphs with no (r + 1)-cliques, T(n,r) has the maximum number of edges.
As a special case, for r = 2, one obtains Mantel's theorem:
The maximum number of edges in an n-vertex triangle-free graph is
In other words, one must delete nearly half of the edges in Kn to obtain a triangle-free graph.
Let 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 that contains no (r + 1)-clique but has more edges than , as follows:
Assume that . Delete vertex and create a copy of vertex (with all of the same neighbors as ); call it . Any clique in the new graph contains at most one vertex among . So this new graph does not contain any (r + 1)-clique. However, it contains more edges:
Case 2: and
Delete vertices and and create two new copies of vertex . Again, the new graph does not contain any (r + 1)-clique. However it contains more edges: .
This proves Claim 1.
The claim proves that one can partition the vertices of into equivalence classes based on their nonneighbors; i.e. two vertices are in the same equivalence class if they are nonadjacent. This implies that is a complete multipartite graph (where the parts are the equivalence classes).
If G is a complete k-partite graph with parts A and B and , then we can increase the number of edges in G by moving a vertex from part A to part B. By moving a vertex from part A to part B, the graph loses edges, but gains edges. Thus, it gains at least edge. This proves Claim 2.
This proof shows that the Turan graph has the maximum number of edges. Additionally, the proof shows that the Turan graph is the only graph that has the maximum number of edges.