Abstract. Dirac’s theorem states that any [Formula: see text]-vertex graph [Formula: see text] with even integer [Formula: see text] satisfying [Formula: see text] contains a perfect matching. We generalize this to [Formula: see text]-uniform linear hypergraphs by proving the following. Any [Formula: see text]-vertex [Formula: see text]-uniform linear hypergraph [Formula: see text] with minimum degree at least [Formula: see text] contains a matching that covers at least [Formula: see text] vertices. This minimum degree condition is asymptotically tight, and obtaining a perfect matching is impossible with any degree condition. Furthermore, we show that if [Formula: see text], then [Formula: see text] contains almost spanning linear cycles, almost spanning hypertrees with [Formula: see text] leaves, and “long subdivisions” of any [Formula: see text]-vertex graphs.

Abstract. Dirac’s theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles. In this paper, we consider another natural generalization of perfect matchings, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices, it is a natural high-dimensional analogue of a perfect matching in graphs. We prove that for sufficiently large integer [Formula: see text] with [Formula: see text], any [Formula: see text]-vertex 3-uniform hypergraph [Formula: see text] with minimum codegree at least [Formula: see text] contains a Steiner triple system. In fact, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs. We conjecture that the number [Formula: see text] can be replaced with [Formula: see text], which would provide an asymptotically tight high-dimensional generalization of Dirac’s theorem.