Constructing Dense Grid-Free Linear $3$-Graphs

Abstract

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 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.