A Simple Counting Argument for Dense Linear Hypergraphs

In connection to the Brown-Erdős-Sós conjecture, we give a short local averaging proof of a density theorem for linear uniform hypergraphs. Let $r\ge 3$, $k\ge 3$, and suppose that $n\ge (r-2)(k-2)+1$. If $H$ is a linear $r$-uniform hypergraph on $n$ vertices and $$\mathrm{\mid }E(H)\mathrm{\mid }\ge \frac{k-2}{r^2((r-2)(k-2)+1)}{n}^2+\frac{n}{r},$$ then $H$ contains $k$ edges spanning at most $(r-2)k+3$ vertices. In the standard linear-density normalization, this gives the asymptotic density threshold $c\ge \frac{r-1}{r}\cdot \frac{k-2}{(r-2)(k-2)+1}+o(1)$. In particular, this yields a simple proof of the large-uniformity form of the Brown-Erdős-Sós theorem, due to Keevash and Long. In the case of triple systems, our bound becomes $c\ge \frac{2(k-2)}{3(k-1)}+o(1)$, improving upon a bound of $\frac{4}{5}$ due to Santos and Tyomkyn.