  The online matching problem was introduced by Karp, Vazirani and Vazirani
(STOC 1990) on bipartite graphs with vertex arrivals. It is well-known that the
optimal competitive ratio is $1-1/e$ for both integral and fractional versions
of the problem. Since then, there has been considerable effort to find optimal
competitive ratios for other related settings. In this work, we go beyond the
graph case and study the online matching problem on $k$-uniform hypergraphs.
For $k=3$, we provide an optimal primal-dual fractional algorithm, which
achieves a competitive ratio of $(e-1)/(e+1)\approx 0.4621$. As our main
technical contribution, we present a carefully constructed adversarial
instance, which shows that this ratio is in fact optimal. It combines ideas
from known hard instances for bipartite graphs under the edge-arrival and
vertex-arrival models. For $k\geq 3$, we give a simple integral algorithm which
performs better than greedy when the online nodes have bounded degree. As a
corollary, it achieves the optimal competitive ratio of 1/2 on 3-uniform
hypergraphs when every online node has degree at most 2. This is because the
special case where every online node has degree 1 is equivalent to the
edge-arrival model on graphs, for which an upper bound of 1/2 is known.
