{"paper":{"title":"When does a tree activate the random graph?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.PR"],"primary_cat":"math.CO","authors_text":"Asaf Cohen Antonir, Asaf Shapira, Maksim Zhukovskii, Mykhaylo Tyomkyn, Yuval Peled","submitted_at":"2025-07-08T06:09:01Z","abstract_excerpt":"Let $F$ and $G$ be two graphs. A spanning subgraph $H$ of $G$ is called weakly $F$-saturated if one can add to $H$ the edges of $G \\setminus H$ in some order, so that whenever a new edge is added, a new copy of $F$ is formed. Obtaining lower bounds for the minimum size $\\mathrm{wsat}(G,F)$ of such an $H$ is a classical problem in extremal combinatorics. In particular, in the past 40 years, various algebraic tools have been developed to prove lower bounds on the weak saturation number $\\mathrm{wsat}(G,F)$. Our paper uncovers a new connection of weak saturation to topology of clique complexes, t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.05697","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/2507.05697/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"}