  In this paper we consider $j$-tuple-connected components in random
$k$-uniform hypergraphs (the $j$-tuple-connectedness relation can be defined by
letting two $j$-sets be connected if they lie in a common edge and consider the
transitive closure; the case $j=1$ corresponds to the common notion of
vertex-connectedness). We determine that the existence of a $j$-tuple-connected
component containing $\Theta (n^j)$ $j$-sets in random $k$-uniform hypergraphs
undergoes a phase transition and show that the threshold occurs at edge
probability $\tfrac{(k-j)!}{\binom{k}{j}-1}n^{j-k}$. Our proof extends the
recent short proof for the graph case by Krivelevich and Sudakov which makes
use of a depth-first search to reveal the edges of a random graph.
  Our main original contribution is a "bounded degree lemma" which controls the
structure of the component grown in the search process.
