Abstract 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 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) ) n ₁{1-p}n, 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.
Diskin et al. (Tue,) studied this question.