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A note on the acquaintance time of random graphs

In this short note, we prove the conjecture of Benjamini, Shinkar, and Tsur on the acquaintance time $AC(G)$ of a random graph $G \in G(n,p)$. It is shown that asymptotically almost surely $AC(G) = O(\log n / p)$ for $G \in G(n,p)$, provided that $pn > (1+ε) \log n$ for some $ε> 0$ (slightly above the threshold for connectivity). Moreover, we show a matching lower bound for dense random graphs, which also implies that asymptotically almost surely $K_n$ cannot be covered with $o(\log n / p)$ copies of a random graph $G \in G(n,p)$, provided that $pn > n^{1/2+ε}$ and $p < 1-ε$ for some $ε>0$. We conclude the paper with a small improvement on the general upper bound showing that for any $n$-vertex graph $G$, we have $AC(G) = O(n^2/\log n)$.