  A graph $H$ is $K_s$-saturated if it is a maximal $K_s$-free graph, i.e., $H$
contains no clique on $s$ vertices, but the addition of any missing edge
creates one. The minimum number of edges in a $K_s$-saturated graph was
determined over 50 years ago by Zykov and independently by Erd\H{o}s, Hajnal
and Moon. In this paper, we study the random analog of this problem: minimizing
the number of edges in a maximal $K_s$-free subgraph of the Erd\H{o}s-R\'enyi
random graph $G(n,p)$. We give asymptotically tight estimates on this minimum,
and also provide exact bounds for the related notion of weak saturation in
random graphs. Our results reveal some surprising behavior of these parameters.
