Constructing Dense Grid-Free Linear $3$-Graphs

TL;DR

This paper constructs dense linear 3-uniform hypergraphs with quadratic edges that avoid certain grid configurations, advancing conjectures in combinatorics and connecting to other fundamental problems.

Contribution

It demonstrates the existence of dense grid-free linear 3-graphs with quadratic edges, making progress on a longstanding conjecture.

Findings

Existence of linear 3-graphs with  vertices and ^2 edges avoiding 3x3 grids

Progress on F1redi and Ruszink1's conjecture

Connections to lower bounds in Brown-Erd1s-S1s problem

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 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.