{"paper":{"title":"Packing and finding paths in sparse random graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Julien Portier, Leo Versteegen, Vesna Ir\\v{s}i\\v{c}","submitted_at":"2024-09-04T15:31:23Z","abstract_excerpt":"Let $G\\sim G(n,p)$ be a (hidden) Erd\\H{o}s-R\\'enyi random graph with $p=(1+ \\varepsilon)/n$ for some fixed constant $ \\varepsilon >0$. Ferber, Krivelevich, Sudakov, and Vieira showed that to reveal a path of length $\\ell=\\Omega\\left(\\frac{\\log(1/ \\varepsilon)}{ \\varepsilon}\\right)$ in $G$ with high probability, one must query the adjacency of $\\Omega\\left(\\frac{\\ell}{p \\varepsilon\\log(1/ \\varepsilon)}\\right)$ pairs of vertices in $G$, where each query may depend on the outcome of all previous queries. Their result is tight up to the factor of $\\log(1/ \\varepsilon)$ in both $\\ell$ and the numbe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.02812","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.02812/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}