{"paper":{"title":"A note on the acquaintance time of random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D. Mitsche, P. Pralat, W. Kinnersley","submitted_at":"2013-05-07T23:05:25Z","abstract_excerpt":"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 $pn > (1+\\epsilon) \\log n$ for some $\\epsilon > 0$ (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 $pn > n^{1/2+\\epsilon}$ and $p < 1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.1675","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}