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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2410.06421","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-10-08T23:20:40Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"271d22336565ff6adbfc803be43d8eb0239a8cc6dbb643c35ef46133727d7f5f","abstract_canon_sha256":"d08c82e2d3485ce56be01438a6a0ab87c6bef69fc6341c2b4839b0b49720d095"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:18:01.323101Z","signature_b64":"AxbQJdLbX+OSPizHTRZq3GpDXGZUAnxqYQ4FM8alIpu1OcHeysm6G3TyIqtDdzf1YA5Nq7Sj4t1zyOV/FZvTAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b45fdb18c0d3249d98fcea68e97dbf304e21d336ffdda25e9c76bee9a80faa47","last_reissued_at":"2026-07-05T09:18:01.322699Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:18:01.322699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mathcal{C}_H(K_n)$. 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