The size of the giant high-order component in random hypergraphs

The phase transition in the size of the giant component in random graphs is one of the most well-studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j-sets (sets of j vertices) are j-connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j-connected if all j-sets are pairwise j-connected. In this paper, we determine the asymptotic size of the unique giant j-connected component in random k-uniform hypergraphs for any k3 and 1j <= k-1.