Saturation in Random Hypergraphs

Let $K<^>r_n$ be the complete $r$ -uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n] \, :\! = \{1,2,\ldots ,n\}$ and whose edge set is $\binom {[n]}{r}$ . We form $G<^>r(n,p)$ by retaining each edge of $K<^>r_n$ independently with probability $p$ . An $r$ -uniform hypergraph $H\subseteq G$ 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 ${\textit {sat}}(G,F)$ , and in a weakly $F$ -saturated hypergraph in $G$ by $\mathop {\mbox{$w$-${sat}$}}\! (G,F)$ . In 2017, Kor & aacute;ndi and Sudakov initiated the study of saturation in random graphs, showing that for constant $p$ , with high probability ${\textit {sat}}(G(n,p),K_s)=(1+o(1))n\log _{\frac {1}{1-p}}n$ , and $\mathop {\mbox{$w$-${sat}$}}\! (G(n,p),K_s)=\mathop {\mbox{$w$-${sat}$}}\! (K_n,K_s)$ . Generalising their results, in this paper, we solve the saturation problem for random hypergraphs $G<^>r(n,p)$ for cliques $K_s<^>r$ , for every $2\le r \lt s$ and constant $p$ .