Embedding loose spanning trees in 3-uniform hypergraphs

In 1995, Komlós, Sárközy and Szemerédi showed that every large $n$-vertex graph with minimum degree at least $(1\mathrm{/}2+\gamma )n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all $\gamma$ and $\mathrm{\Delta }$, and $n$ large, every $n$-vertex 3-uniform hypergraph of minimum vertex degree $(5\mathrm{/}9+\gamma )\left(\genfrac{}{}{0px}{}{n}{2}\right)$ contains every loose spanning tree $T$ with maximum vertex degree $\mathrm{\Delta }$. This bound is asymptotically tight, since some loose trees contain perfect matchings.