Embedding loose trees in $k$-uniform hypergraphs

Abstract

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi shows that every large $n$-vertex graph with minimum degree at least $(1/2+\gamma)n$ contains all spanning trees of bounded degree. We generalised this result to loose spanning hypertrees in $k$-uniform hypergraphs, that is, linear hypergraphs obtained by subsequently adding edges sharing a single vertex with a previous edge. We give a general sufficient condition for embedding loose trees with bounded degree. In particular, we show that for all $k\ge 4$, every $n$-vertex $k$-uniform hypergraph with $n\ge n_0(k,\gamma, \Delta)$ and minimum $(k-2)$-degree at least $(1/2+\gamma)\binom{n}{k-2}$ contains every spanning loose tree with maximum vertex degree at most $\Delta$. This bound is asymptotically tight. This generalises a result of Pehova and Petrova, who proved the case when $k=3$ and of Pavez-Sign\'e, Sanhueza-Matamala and Stein, who considered the codegree threshold for bounded degree tight trees.