  Benjamini, Shinkar, and Tsur stated the following conjecture on the
acquaintance time: asymptotically almost surely $AC(G) \le p^{-1} \log^{O(1)}
n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently,
Kinnersley, Mitsche, and the second author made a major step towards this
conjecture by showing that asymptotically almost surely $AC(G) = O(\log n /
p)$, provided that $G$ has a Hamiltonian cycle. In this paper, we finish the
task by showing that the conjecture holds in the strongest possible sense, that
is, it holds right at the time the random graph process creates a connected
graph. Moreover, we generalize and investigate the problem for random
hypergraphs.
