Grid-free linear hypergraphs via Cayley-Bacharach

TL;DR

This paper introduces a new construction of linear hypergraphs that avoids containing a specific grid pattern, expanding understanding of hypergraph structures and their extremal properties.

Contribution

It provides a novel construction for r-uniform linear hypergraphs with many edges that do not contain an r-by-r grid, for all r ≥ 3.

Findings

Constructs hypergraphs with Θ(n^2) edges avoiding r-by-r grids

Extends previous results for r ≥ 4 to all r ≥ 3

Offers new insights into hypergraph extremal configurations

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 F\"uredi--Ruszink\'o, Glock--Joos--Kim--K\"uhn--Lichev, Delcourt--Postle for $r \geq 4$, as well as the subsequent constructions of Gishboliner--Shapira and Solymosi for the case $r=3$.