Let $G(V,E)$ be a **connected**, simple and undirected graph with the additional constraint, that each edge is contained in the same number $k_T$ of triangles; i.e. that $G$ is regular w.r.t. to that number of triangles.

Question:

what are non-trivial bounds on the size of the maximal clique of $k_T$-regularconnectedgraphs with $n$ vertices as a function of $k_T$ and $n$?

for $k_T=2$ the minimal clique-size is $4$ iff $n=4$ and $3$ iff $n>4$ and the question amounts to whether triangle-regularity allows for sharper estimates of the maximal clique size.