The quadratic Brown--Erd\H{o}s--S\'os problem for 3-uniform hypergraphs with 8 and 9 edges

Abstract

The famous and actively studied problem of Brown--Erd\H{o}s--S\'os from 1973 asks for $f^{(r)}(n;s,k)$, the maximum number of edges in an $r$-graph with $n$ vertices in which no $s$ vertices span $k$ or more edges. In this paper, we concentrate on the case $r=3$ and $s=k+2$, with $k\ge2$ fixed and $n\to\infty$; then it is easy to show that the extremal function grows quadratically in $n$. Delcourt and Postle proved that the limit $\pi(k):=\lim_{n\to\infty} f^{(3)}(n;k+2,k)/n^2$ exists for every $k$. While Brown, Erd\H{o}s and S\'os observed that $\pi(2)=1/6$ already in the 1970s, the value of $\pi(k)$ for $3\le k\le 7$ was determined only recently (by various subgroups of Glock, Joos, Kim, K\"uhn, Lichev, Pikhurko, and Sun). Very recently, Chao, Huang and Liu determined $\pi(k)$ for every odd $k$. Independently of the last result, we show that $\pi(9)=1/5$. Also, we prove that $\pi(8)\le {5053}/{26544}$, which is within $0.0029$ of the best known lower bound $\pi(8)\ge 3/16$. The new upper bounds are obtained by expressing some previous arguments as a linear program and then using a computer to generate and solve its instances. Our proof of the lower bound on $\pi(9)$ is based on a finite field construction combined with existing packing results.