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

Abstract

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 $n\geq n_0$ . We prove this conjecture.