The size of the giant component in random hypergraphs: A short proof

We consider connected components in k-uniform hypergraphs for the following notion of connectedness: given integers k≥2 and 1≤j≤k−1, two j-sets (of vertices) lie in the same j-component if there is a sequence of edges from one to the other such that consecutive edges intersect in at least j vertices.
We prove that certain collections of j-sets constructed during a breadth-first search process on j-components in a random k-uniform hypergraph are reasonably regularly distributed with high probability. We use this property to provide a short proof of the asymptotic size of the giant j-component shortly after it appears.