Dirac's theorem states that if in a graph G on n vertices
the degree of every vertex is at least n/2 then the graph
contains a Hamiltonian cycle.

In a 3-uniform hypergraph H, the co-degree of two vertices
x, y, denoted by d(x,y)  is the number of vertices z such
that {x,y,z} is an edge in H.

We are going to prove that if d(x, y) is at least n/2 for
every x and y, then H contains a Hamiltonian cycle, meaning
a cyclical ordering of the vertices such that every three
consecutive vertices form an edge. We will also prove the
analogous statement for k-uniform hypergraphs with k>3.

This is joint work with R. Rucinski and V. Rodl.