= Triangle support line between bipartite and tripartite Turan graphs
For $n\geq3$, 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> satisfies
$$
e(G)-c_n k_3(G)\leq e_2.
$$
Thus 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 $c_n\geq2/n$ 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.
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