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 $\delta >0$ and integer $k\ge 3$ every sufficiently large linear $3$-uniform hypergraph of size $\delta n^2$ contains some $k$ edges spanning at most $k+3$ vertices. We prove it to hold for $\delta >4\mathrm{/}5$, establishing the first bound of this kind.