Saturation in Random Hypergraphs

Let Kʳₙ be the complete r-uniform hypergraph on n vertices, that is, the hypergraph whose vertex set is [n]:=\1,2,...,n\ and whose edge set is [n]r. We form Gʳ(n,p) by retaining each edge of Kʳₙ independently with probability p. An r-uniform hypergraph H⊆ G is F-saturated if H does not contain any copy of F, but any missing edge of H in G creates a copy of F. Furthermore, we say that H is weakly F-saturated in G if H does not contain any copy of F, but the missing edges of H in G can be added back one-by-one, in some order, such that every edge creates a new copy of F. The smallest number of edges in an F-saturated hypergraph in G is denoted by $sat(G,F)$, and in a weakly F-saturated hypergraph in G by $wsat(G,F)$. In 2017, Kor\'andi and Sudakov initiated the study of saturation in random graphs, showing that for constant p, with high probability sat(G(n,p),Kₛ)=(1+o(1))nlog1/1-pn, and wsat(G(n,p),Kₛ)=wsat(Kₙ,Kₛ). Generalising their results, in this paper, we solve the suturation problem for random hypergraphs for every 2≤ r < s and constant p.