Perfect matching in high-degree hypergraphs

In graph theory, perfect matching in high-degree hypergraphs is a research avenue trying to find sufficient conditions for existence of a perfect matching in a hypergraph, based only on the degree of vertices or subsets of them.

Khan's conditions: smallest vertex degree

Given a hypergraph H = (V,E), the degree of a vertex v in V is the number of hyperedges containing v.

Let H = (V, E) be a 3-uniform hypergraph with on n=3 k vertices, such that every the degree of each vertex is at least ${n-1 \choose 2}-{2n/3 \choose 2}+1$. Then H contains a perfect matching - matching of size k. This result is the best possible.

Let H = (V, E) be a 4-uniform hypergraph with on n = 4 k vertices, such that the degree of each vertex is at least ${n-1 \choose 3}-{3n/4 \choose 3}$. Then H contains a perfect matching - matching of size k. This result is the best possible.

These results generalize a known result on graphs (which are 2-uniform hypergraphs): if the degree of each vertex in a graph is at least ${n-1 \choose 1}-{n/2 \choose 1}+1=n/2$, then the graph admits a perfect matching. This follows from Dirac's theorem on Hamiltonian cycles, which says that if the degree of each vertex is at least n/2, then the graph admits a Hamiltonian cycle. The n/2 is tight, since the complete bipartite graph on (n/2-1, n/2+1) vertices has degree n/2-1 but does not admit a perfect matching.