On the quadratic 8-edge case of the Brown-Erd\H{o}s-S\'os problem

TL;DR

This paper determines the exact limit of the maximum number of edges in large hypergraphs avoiding certain small edge configurations for the case k=8, extending previous results for smaller k.

Contribution

It computes the limit for the case k=8 for all uniformities r≥4 and provides a conjectured sharp lower bound for r=3, advancing understanding of the Brown-Erdős-Sós problem.

Findings

Limit value determined for k=8 and r≥4.

Lower bound established for r=3, conjectured to be sharp.

Extends known results to the case k=8.

Abstract

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ edges on at most $s$ vertices. Brown, Erd\H{o}s and S\'os conjectured in 1973 that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. Recently, Delcourt and Postle settled the conjecture and their approach was generalised by Shangguan to every uniformity $r\ge 4$: the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ exists for all $r\ge 3$ and $k\ge 2$. The value of the limit is currently known for $k\in \{2,3,4,5,6,7\}$ due to various results authored by Glock, Joos, Kim, K\"{u}hn, Lichev, Pikhurko, R\"odl and Sun. In this paper we consider the case $k=8$, determining the value of the limit for each $r\ge 4$ and presenting a lower bound for $k=3$ that we conjecture to be sharp.