  We consider the problem of inferring a matching hidden in a weighted random
$k$-hypergraph. We assume that the hyperedges' weights are random and
distributed according to two different densities conditioning on the fact that
they belong to the hidden matching, or not. We show that, for $k>2$ and in the
large graph size limit, an algorithmic first order transition in the signal
strength separates a regime in which a complete recovery of the hidden matching
is feasible from a regime in which partial recovery is possible. This is in
contrast to the $k=2$ case where the transition is known to be continuous.
Finally, we consider the case of graphs presenting a mixture of edges and
$3$-hyperedges, interpolating between the $k=2$ and the $k=3$ cases, and we
study how the transition changes from continuous to first order by tuning the
relative amount of edges and hyperedges.
