{"paper":{"title":"Universality in random graphs via optimal linking systems: trees and beyond","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Asaf Cohen Antonir, Lyuben Lichev, Maksim Zhukovskii","submitted_at":"2026-08-06T17:55:25Z","abstract_excerpt":"We develop a framework for proving universality results in sparse random graphs. As a first application, we show that there exists an absolute constant $C>1$ such that, with high probability, for every fixed constant $\\Delta$, the binomial random graph $G(n,C\\ln n/n)$ contains every $n$-vertex tree with maximum degree at most $\\Delta$. This answers a question of Montgomery (Advances in Mathematics, 2019). We also determine, for every $p$ satisfying $C\\ln n/n\\leq p=n^{-1+o(1)}$, the minimum girth $\\ell$ (up to an absolute multiplicative constant) for which with high probability $G(n,p)$ contain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06358","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/2608.06358/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"}