Dirac's theorem for linear hypergraphs

Dirac's theorem states that any n-vertex graph G with even integer n satisfying (G) n/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 nk + (1) contains a matching that covers at least (1-o (1) ) n vertices. This minimum degree condition is asymptotically tight and obtaining perfect matching is impossible with any degree condition. Furthermore, we show that if (H) (1k+o (1) ) n, then H contains almost spanning linear cycles, almost spanning hypertrees with o (n) leaves, and ``long subdivisions'' of any o (n) -vertex graphs.