Grid-free linear hypergraphs via Cayley-Bacharach

Abstract

We give a new construction showing that for every $r\ge 3$, there exists an $r$-uniform linear hypergraph on $n$ vertices with ${\Theta }_r(n^2)$ edges and no copy of the $r\times r$ grid. This complements the works of Füredi--Ruszinkó, Glock--Joos--Kim--Kühn--Lichev, Delcourt--Postle for $r\ge 4$, as well as the subsequent constructions of Gishboliner--Shapira and Solymosi for the case $r=3$.

11 pages