High-order Phase Transition in Random Hypergrpahs

In this paper, we study the high-order phase transition in random r-uniform hypergraphs. For a positive integer n and a real p ∈ [0, 1], let H := H(n, p) be the random r-uniform hypergraph on the vertex set [n], where each r-set is an edge with probability p independently. For 1 ≤ s ≤ r − 1 and two s-sets S and S, we say S is connected to S if there is a sequence of alternating s-sets and edges S0, F1, S1, F2, . . . , Fk, Sk which satisfies S0, S1, . . . , Sk are s-sets, S0 = S, Sk = S , F1, F2, . . . , Fk are edges of H , and Si−1 ∪ Si ⊂ Fi for each 1 ≤ i ≤ k. This is an equivalence relation over the family of all s-sets ( [n] s ) and results a partition: ( V s ) = ∪iCi. Each Ci is called an s-th-order connected component and a component Ci is a giant s-th-order connected component if |Ci| = Θ(n ). We prove that the sharp threshold of the existence of the s-th-order giant connected components in H(n, p) is 1+o(1) (