OPEN

Let $G$ be a connected graph with $n$ vertices, minimal degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\mid d$ then
\[D\leq \frac{2(r-1)(3r+2)}{(2r^2-1)d}n+O(1),\]
and if $G$ contains no $K_{2r+1}$ and $3r-1 \mid d$ then
\[D\leq \frac{3r-1}{rd}n+O(1).\]

A problem of Erdős, Pach, Pollack, and Tuza.