{"paper":{"title":"Random embeddings of bounded degree trees with optimal spread","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alp M\\\"uyesser, Cl\\'ement Legrand-Duchesne, Paul Bastide","submitted_at":"2024-09-10T16:55:15Z","abstract_excerpt":"A seminal result of Koml\\'os, S\\'ark\\\"ozy, and Szemer\\'edi states that any n-vertex graph G with minimum degree at least (1/2 + {\\alpha})n contains every n-vertex tree T of bounded degree. Recently, Pham, Sah, Sawhney, and Simkin extended this result to show that such graphs G in fact support an optimally spread distribution on copies of a given T, which implies, using the recent breakthroughs on the Kahn-Kalai conjecture, the robustness result that T is a subgraph of sparse random subgraphs of G as well. Pham, Sah, Sawhney, and Simkin construct their optimally spread distribution by following"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.06640","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/2409.06640/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"}