  In the online hypergraph matching problem, hyperedges of size $k$ over a
common ground set arrive online in adversarial order. The goal is to obtain a
maximum matching (disjoint set of hyperedges). A na\"ive greedy algorithm for
this problem achieves a competitive ratio of $\frac{1}{k}$. We show that no
(randomized) online algorithm has competitive ratio better than
$\frac{2+o(1)}{k}$. If edges are allowed to be assigned fractionally, we give a
deterministic online algorithm with competitive ratio $\frac{1-o(1)}{\ln(k)}$
and show that no online algorithm can have competitive ratio strictly better
than $\frac{1+o(1)}{\ln(k)}$. Lastly, we give a $\frac{1-o(1)}{\ln(k)}$
competitive algorithm for the fractional edge-weighted version of the problem
under a free disposal assumption.
