The quadratic Brown--Erdős--Sós problem for 3-uniform hypergraphs with 8 and 9 edges

The famous and actively studied problem of Brown--Erdős--Sós from 1973 asks for $f^{(r)}(n;s,k)$, the maximum number of edges in an $r$-graph with $n$ vertices in which no $s$ vertices span $k$ or more edges. In this paper, we concentrate on the case $r=3$ and $s=k+2$, with $k\ge2$ fixed and $n\to