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))nlog11−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 Krs , for every 2≤r<s and constant p .