Brown–Erdős–Sós conjecture for dense linear triple systems

Let $k\geq 4$ and $0<\delta\leq 1$. A linear $3$-graph is a $3$-uniform hypergraph in which any two edges share at most one vertex; for a linear $3$-graph $\mathcal{H}$ with $n$ vertices and $m$ edges, its linear density is

$$d^{\operatorname{lin}}(\mathcal{H})=\frac{3m}{\binom{n}{2}}.$$

A $(k+3,k)$-configuration is a set of $k$ edges whose union contains at most $k+3$ vertices. Brown–Erdős–Sós conjecture for dense linear triple systems. For every $k\geq 4$ and $0<\delta\leq 1$ there exists $n_0=n_0(k,\delta)$ such that every linear $3$-graph $\mathcal{H}$ with $n\geq n_0$ vertices and $d^{\operatorname{lin}}(\mathcal{H})\geq\delta$ contains a $(k+3,k)$-configuration. The general Brown–Erdős–Sós conjecture is known in the case $k=3$ by the Ruzsa–Szemerédi $(6,3)$-theorem, while the cases $k\geq 4$ remain open; the paper proves the assertion for densities $\delta>4/5$.