{"paper":{"title":"On the $H$-space of a random graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Jeff Kahn, Quentin Dubroff","submitted_at":"2024-10-08T23:20:40Z","abstract_excerpt":"The edge space $\\mathcal{E}(G)$ of a graph $G$ is the vector space $\\mathbb{F}_2^{E(G)}$ with members naturally identified with subgraphs of $G$, and the $H$-space is the subspace $\\mathcal{C}_H(G)$ of $ \\mathcal{E}(G)$ spanned by copies of the graph $H$. We are interested in when the random graph $G = G_{n,p}$ is likely to satisfy \\[\\mathcal{C}_H(G) = \\mathcal{W}_H(G),\\] where $\\mathcal{W}_H(G)$ takes one of four natural values, depending on the value of $\\mathcal{C}_H(K_n)$. We show that for strictly $2$-balanced $H$, w.h.p. the above equality holds whenever every edge of $G$ is in a copy of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.06421","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/2410.06421/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"}