  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 the 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 $n$ with $n
\equiv 1 \text{ or } 3 \pmod{6},$ any $n$-vertex $3$-uniform hypergraph $H$
with minimum codegree at least $\left(\frac{3 + \sqrt{57}}{12} + o(1) \right)n
= (0.879... + o(1))n$ 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 $\frac{3 +
\sqrt{57}}{12}$ can be replaced with $\frac{3}{4}$ which would provide an
asymptotically tight high-dimensional generalization of Dirac's theorem.
