{"paper":{"title":"Cycle lengths in the percolated hypercube","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Joshua Erde, Lyuben Lichev, Michael Anastos, Michael Krivelevich, Mihyun Kang, Sahar Diskin","submitted_at":"2025-06-20T09:15:22Z","abstract_excerpt":"Let $Q^d_p$ be the random subgraph of the $d$-dimensional binary hypercube obtained after edge-percolation with probability $p$. It was shown recently by the authors that, for every $\\varepsilon > 0$, there is some $c = c(\\varepsilon)>0$ such that, if $pd\\ge c$, then typically $Q^d_p$ contains a cycle of length at least $(1-\\varepsilon)2^d$. We strengthen this result to show that, under the same assumptions, typically $Q^d_p$ contains cycles of all even lengths between $4$ and $(1-\\varepsilon)2^d$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.16858","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/2506.16858/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"}