Dirac's theorem for linear hypergraphs

Dirac’s theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $\delta (G)\ge n\mathrm{/}2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex $k$-uniform linear hypergraph $H$ with minimum degree at least $\frac{n}{k}+\mathrm{\Omega }(1)$ contains a matching that covers at least $(1-o(1))n$ vertices. This minimum degree condition is asymptotically tight and obtaining a perfect matching is impossible with any degree condition. Furthermore, we show that if $\delta (H)\ge (\frac{1}{k}+o(1))n$, then $H$ contains almost spanning linear cycles, almost spanning hypertrees with $o(n)$ leaves, and ``long subdivisions’’ of any $o(\sqrt{n})$-vertex graphs.