Turan theorem

Among $n$-vertex graphs containing no $K_{r+1}$, the maximum number of edges is attained by the complete $r$-partite graph whose part sizes differ by at most one, denoted $T_r(n)$.

A clique-free graph near the Turan theorem edge bound is close to a complete multipartite graph. Choose a maximum-degree vertex, split its neighbourhood from the remaining vertices, and apply induction inside the neighbourhood. The degree bound implies that missing cross edges are at least twice the number of edges inside the remaining class. This inequality controls all three types of edits and gives the constant three.

Fix an odd cycle length. If the graph has edge density at least $1/2-ϵ$ and sufficiently few copies of that cycle, odd-cycle copies from positive triangle density forces few triangles in a graph. The triangle removal lemma deletes at most $ϵ\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ edges, after which the clique-free edit bound applies with two parts. Absorbing its linear rounding error into the density margin gives the displayed bound for large order.

The Turan graph $T_r(n)$ is the complete multipartite graph on $n$ vertices with $r$ parts whose sizes differ by at most one.

For $n\ge 3$, put $e_i=e(T_i(n))$, $t_3=k_3(T_3(n))$, and $c_n=(e_3-e_2)/t_3$. Every graph on $n$ vertices satisfiesThus an edge count $e_2+\theta (e_3-e_2)$ forces at least $\theta t_3$ triangles in a graph. This is an exact finite bound, with the rounding in the Turan graph retained. To prove it, use edge-triangle symmetrization. The inequality $c_n\ge 2/n$ allows merging the two smallest classes when there are at least four classes. With three classes, fixing the middle class makes the objective affine in the product of the other two sizes; either merge them or balance them. Only $T_2(n)$ and $T_3(n)$ need remain.

Zykov symmetrization replaces one of two nonadjacent vertices by a clone of the other. Cloning the vertex of larger degree does not decrease the edge count and does not create a larger clique. Iteration reduces the extremal problem for clique-free graphs to complete multipartite graphs.

For each real $c$, some maximizer of $e(G)-c{k}_3(G)$ among graphs on $n$ vertices is complete multipartite graph. Among maximizers maximize ${\sum }_{v}d(v{)}^2$. Cloning either of two nonadjacent vertices preserves the objective, since maximality forces their local contributions $d(v)-ce(G[N(v)])$ to agree. If their vertex neighbourhoods differ, the sum of the two changes in the squared-degree of a vertex objective is twice the size of their symmetric difference, a contradiction. Nonadjacency therefore partitions the vertices into classes with identical vertex neighbourhoods.

Every $n$-vertex graph containing no $K_{r+1}$ satisfies

A graph-theoretic rhombus is two triangles sharing an edge. Every rhombus-free graph on $n\ge 4$ vertices has at mostedges. If the graph is triangle-free this is Turán's theorem. Otherwise remove a triangle: every remaining vertex has at most one neighbor in it, and induction bounds the edge count by
$\lfloor (n-3{)}^2/4\rfloor +n\le \lfloor n^2/4\rfloor$.

The triangular prism has six vertices and nine edges. Each edge lies in at most one triangle, so it has no rhombus; its two triangles show that it is not $K_{3,3}=T_2(6)$.