SOLVED
Let $k\geq 4$. Is it true that
\[\mathrm{ex}(n;H_k) \ll_k n^{3/2},\]
where $H_k$ is the graph on vertices $x,y_1,\ldots,y_k,z_1,\ldots,z_{\binom{k}{2}}$, where $x$ is adjacent to all $y_i$ and each pair of $y_i,y_j$ is adjacent to a unique $z_i$.
The answer is yes, proved by Füredi
[Fu91], who proved that
\[\mathrm{ex}(n;H_k) \ll (kn)^{3/2}.\]
This was improved to
\[\mathrm{ex}(n;H_k) \ll kn^{3/2}\]
by Alon, Krivelevich, and Sudakov
[AKS03].
Since each $H_k$ is 2-degenerate this is a special case of [146].
