{"paper":{"title":"Robustness of the Sauer-Spencer Theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Julia B\\\"ottcher, Mihir Neve, Peter Allen, Yoshiharu Kohayakawa","submitted_at":"2025-07-04T15:51:52Z","abstract_excerpt":"We prove a robust version of a graph embedding theorem of Sauer and Spencer. To state this sparser analogue, we define $G(p)$ to be a random subgraph of $G$ obtained by retaining each edge of $G$ independently with probability $p \\in [0,1]$, and let $m_1(H)$ be the maximum $1$-density of a graph $H$. We show that for any constant $\\Delta$ and $\\gamma > 0$, if $G$ is an $n$-vertex host graph with minimum degree $\\delta(G) \\geq (1 - 1/2\\Delta + \\gamma)n$ and $H$ is an $n$-vertex graph with maximum degree $\\Delta(H) \\leq \\Delta$, then for $p \\geq Cn^{-1/m_1(H)}\\log n$, the random subgraph $G(p)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.03676","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/2507.03676/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"}