The Brown-Erdős-Sós conjecture in dense triple systems

The famous Brown-Erdős-Sós conjecture from 1973 states, in an equivalent form, that for any fixed $δ>0$ and integer $k\geq 3$ every sufficiently large linear $3$-uniform hypergraph of size $δn^2$ contains some $k$ edges spanning at most $k+3$ vertices. We prove it to hold for $δ>4/5$, establishing the first bound of this kind.