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 nr. 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) ) n₁₁-₏n, 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.