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 w$-${sat}\! (G,F) . In 2017, Korándi 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))nlog 1/1-pn , and w$-${sat}\! (G(n,p),Kₛ)=w$-${sat}\! (Kₙ,Kₛ) . Generalising their results, in this paper, we solve the saturation problem for random hypergraphs Gʳ(n,p) for cliques Kₛʳ , for every 2≤ r < s and constant p .