  Let $r \ge 2$ be a fixed constant and let $ {\mathcal H}$ be an $r$-uniform,
$D$-regular hypergraph on $N$ vertices. Assume further that $ D \to \infty$ as
$N \to \infty$ and that degrees of pairs of vertices in ${\mathcal H}$ are at
most $L$ where $L \ = D/ (\log N)^{\omega(1)}$. We consider the random greedy
algorithm for forming a matching in $ \mathcal{H}$. We choose a matching at
random by iteratively choosing edges uniformly at random to be in the matching
and deleting all edges that share at least one vertex with a chosen edge before
moving on to the next choice. This process terminates when there are no edges
remaining in the graph. We show that with high probability the proportion of
vertices of $ {\mathcal H}$ that are not saturated by the final matching is at
most $ (L/D)^{ \frac{ 1}{ 2(r-1) } + o(1) } $. This point is a natural barrier
in the analysis of the random greedy hypergraph matching process.

Combinatorics