# Forbidden $K_4$ in a subgraph of $K_{10}(n)$

In terms of $$n\geq 1$$, find the maximum possible number of edges in a subgraph $$H$$ of $$K_{10}(n)$$, the complete $$10$$-partite graph with $$n$$ vertices in each class, containing no copy of $$K_4$$.

It is suggested to apply Turan's theorem. For $$n=1$$ Turan gives $$33$$ (which is attainable, as $$K_{10}(1) = K_{10}$$ gives on restrictions on the edges of $$H$$). In general, Turan gives the bound $$\lfloor \frac{100n^2}{3} \rfloor$$ but is this always attainable, i.e. can we find $$T_3(10n)$$ in $$K_{10}(n)$$? I am not even sure this is always possible.

• Yes, in order to find in $K_{10}(n)$ Turán’s graph with $10n$ vertices and $3$ partite sets when $n$ is not divisible by $3$ we have to split one of partite set of $K_{10}(n)$ to different partite set of Turán’s graph, but then vertices from different parts of the split set cannot be adjacent to each other. – Alex Ravsky Apr 26 at 8:41