Asymptotics of the Brown--Erd\H{o}s--S\'os problem at integer exponents

Abstract

The Brown--Erd\H{o}s--S\'os problem is a fundamental problem in sparse hypergraph Tur\'an theory. For integers $r,k\ge 2$ and $s\ge r$, the problem asks for the maximum number $f^{(r)}(n;s,k)$ of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ distinct edges spanning at most $s$ vertices. In 1971, Brown, Erd\H{o}s, and S\'os proved that $f^{(r)}\bigl(n;(r-t)k+t,k\bigr)=\Theta(n^t)$ for all $r>t\ge 2$ and $k\ge 2$. However, the existence and the value of the leading coefficient have remained largely open. We determine the coefficient $\pi(r,t,k):=\lim_{n\to\infty}n^{-t}f^{(r)}\bigl(n;(r-t)k+t,k\bigr)$ for every such $r,t,k$, except when $(r,t)=(3,2)$ and $k\ge 4$ is even. In particular, \[ \pi(r,t,k) = \begin{cases} \frac{2}{t!\bigl(2\binom{r}{t}-1\bigr)},&\text{if $k$ is odd},\\ \frac{1}{t!\binom{r}{t}},&\text{if $k$ is even and $r\ge4$}. \end{cases} \] Surprisingly, for $r\ge4$, the limit $\pi(r,t,k)$ depends on $k$ only through its parity, not its value. For the remaining case, we prove that $\pi(3,2,k)>1/6$ for all even $k\ge 4$, showing that the natural packing construction is never asymptotically optimal. As an application, we extend a connection of Bennett, Cushman, and Dudek to arbitrary uniformity to resolve several cases of the Erd\H{o}s--Gy\'arf\'as--Shelah generalized Ramsey problem.