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Turan's theorem tells us that if m is the number of edges in a graph with n vertices and clique number r, then 2m <= (r - 1)n^2/r.

If t denotes the number of triangles, is there a similar upper bound for t, using r and n and/or m for r >= 3? Many thanks.

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Zykov [On some properties of linear complexes,Mat.Sbornik N.S.24(66), 163–188, 1949] proved the following generalisation of Turán’s Theorem:

Theorem: For all integers $k \geq s\geq 0$, the maximum number of $s$-cliques in a graph with $n$ vertices and no $(k + 1)$-clique is $\binom{k}{s}(\frac{n}{k})^s$.

This theorem with $s=3$ and $k=r$ solves your question. An easy proof is in my paper at http://users.monash.edu.au/~davidwo/papers/Wood-GC07.pdf

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