The quadratic Brown--Erdős--Sós problem for 3-uniform hypergraphs with 8 and 9 edges

Abstract: The famous and actively studied problem of Brown--Erdős--Sós from 1973 asks for $f<sup>{(r)}(n;s,k)$, 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 $$π(k):=\lim_{n\to\infty} f<sup>{(3)}(n;k+2,k)/n<sup>2$$ exists for every $k$. While Brown, Erdős and Sós observed that $π(2)=1/6$ already in the 1970s, the value of $π(k)$ for $3\le k\le 7$ was determined only recently (by various subgroups of Glock, Joos, Kim, Kühn, Lichev, Pikhurko, and Sun). Very recently, Chao, Huang and Liu determined $π(k)$ for every odd $k$. Independently of the last result, we show that $π(9)=1/5$. Also, we prove that $π(8)\le {5053}/{26544}$, which is within $0.0029$ of the best known lower bound $π(8)\ge 3/16$. 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 $π(9)$ is based on a finite field construction combined with existing packing results.