  Suppose $k\nmid n$ and $H$ is an $n$-vertex $k$-uniform hypergraph. A near
perfect matching in $H$ is a matching of size $\lfloor n/k\rfloor$. We give a
divisibility barrier construction that prevents the existence of near perfect
matchings in $H$. This generalizes the divisibility barrier for perfect
matchings. We give a conjecture on the minimum $d$-degree threshold forcing a
(near) perfect matching in $H$ which generalizes a well-known conjecture on
perfect matchings. We also verify our conjecture in various cases. Our proof
makes use of the lattice-based absorbing method that the author used recently
to solve two other problems on matching and tilings for hypergraphs.

Combinatorics