A proof of the Elliott-Rödl conjecture on hypertrees in Steiner triple systems

Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and Rödl conjectured that for any given $\mu >0$, there exists $n_0$ such that the following holds. Every $n$-vertex Steiner triple system contains all hypertrees with at most $(1-\mu )n$ vertices whenever $n\ge n_0$. We prove this conjecture.