  We show that there exist linear $3$-uniform hypergraphs with $n$ vertices and
$\Omega(n^2)$ edges which contain no copy of the $3 \times 3$ grid. This makes
significant progress on a conjecture of F\"{u}redi and Ruszink\'{o}. We also
discuss connections to proving lower bounds for the $(9,6)$
Brown-Erd\H{o}s-S\'{o}s problem and to a problem of Solymosi and Solymosi.
