Constructing dense grid-free linear 3-graphs.

We show that there exist linear 3-uniform hypergraphs with n vertices and Ω (n2) edges which contain no copy of the 3 × 3 grid. This makes significant progress on a conjecture of Füredi and Ruszinkó. We also discuss connections to proving lower bounds for the (9,6) Brown-Erdős-Sós problem and to a problem of Solymosi and Solymosi. [ABSTRACT FROM AUTHOR]