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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 atmost $L$ where $L \ = D/ (\log N)^{\omega(1)}$. We consider the random greedyalgorithm for forming a matching in $ \mathcal{H}$. We choose a matching atrandom by iteratively choosing edges uniformly at random to be in the matchingand deleting all edges that share at least one vertex with a chosen edge beforemoving on to the next choice. This process terminates when there are no edgesremaining in the graph. We show that with high probability the proportion ofvertices of $ {\mathcal H}$ that are not saturated by the final matching is atmost $ (L/D)^{ \frac{ 1}{ 2(r-1) } + o(1) } $. This point is a natural barrierin the analysis of the random greedy hypergraph matching process.

A natural barrier in random greedy hypergraph matching