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Let $D_{H}(n)$ denote the maximum possible cardinality of an $H$-code, and let $d_{H}(n)=D_{H}(n)/2^{n \\choose 2}$. Alon observed that a lower bound on $d_{H}(n)$ can be obtained by attaining an upper bound on the number of colors needed to edge-color $K_n$ so that every copy of $H$ has an odd color class.\n  Motivated by this observation, we define $g(G,H)$ to be the minimum number of colors"},"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":"2307.01314","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-07-03T19:40:31Z","cross_cats_sorted":[],"title_canon_sha256":"b9cb3dfde5c709a0df61a9c6cf92692ca3bfb7560b514d1cc72efe95b1c34cc1","abstract_canon_sha256":"3e541a9f5ac4b9b1986cc1dcb39c59873da3a6956c3e978d4c031d1a75742ec6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:43:03.709159Z","signature_b64":"8jZEEnT762SXyyTsPEe6cVNxRuUlM1zbTkdMbs4R72Vt/zxfmcof343ZIQBb7G7JDiQl5quDhAT32Uw8y0/pAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5e22a66bdcfd71298f5ae68e30356ecb55f843f9733b39e14b1371b19d415d0","last_reissued_at":"2026-07-05T06:43:03.708659Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:43:03.708659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Edge-coloring a graph $G$ so that every copy of a graph $H$ has an odd color class","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emily Heath, Patrick Bennett, Shira Zerbib","submitted_at":"2023-07-03T19:40:31Z","abstract_excerpt":"Recently, Alon introduced the notion of an $H$-code for a graph $H$: a collection of graphs on vertex set $[n]$ is an $H$-code if it contains no two members whose symmetric difference is isomorphic to $H$. Let $D_{H}(n)$ denote the maximum possible cardinality of an $H$-code, and let $d_{H}(n)=D_{H}(n)/2^{n \\choose 2}$. Alon observed that a lower bound on $d_{H}(n)$ can be obtained by attaining an upper bound on the number of colors needed to edge-color $K_n$ so that every copy of $H$ has an odd color class.\n  Motivated by this observation, we define $g(G,H)$ to be the minimum number of colors"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.01314","kind":"arxiv","version":3},"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/2307.01314/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2307.01314","created_at":"2026-07-05T06:43:03.708717+00:00"},{"alias_kind":"arxiv_version","alias_value":"2307.01314v3","created_at":"2026-07-05T06:43:03.708717+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.01314","created_at":"2026-07-05T06:43:03.708717+00:00"},{"alias_kind":"pith_short_12","alias_value":"2XRCUZV5Z7LR","created_at":"2026-07-05T06:43:03.708717+00:00"},{"alias_kind":"pith_short_16","alias_value":"2XRCUZV5Z7LRFGHV","created_at":"2026-07-05T06:43:03.708717+00:00"},{"alias_kind":"pith_short_8","alias_value":"2XRCUZV5","created_at":"2026-07-05T06:43:03.708717+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.07322","citing_title":"New results on the odd- and unique-Ramsey numbers","ref_index":6,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2XRCUZV5Z7LRFGHVVZUOGA2W5S","json":"https://pith.science/pith/2XRCUZV5Z7LRFGHVVZUOGA2W5S.json","graph_json":"https://pith.science/api/pith-number/2XRCUZV5Z7LRFGHVVZUOGA2W5S/graph.json","events_json":"https://pith.science/api/pith-number/2XRCUZV5Z7LRFGHVVZUOGA2W5S/events.json","paper":"https://pith.science/paper/2XRCUZV5"},"agent_actions":{"view_html":"https://pith.science/pith/2XRCUZV5Z7LRFGHVVZUOGA2W5S","download_json":"https://pith.science/pith/2XRCUZV5Z7LRFGHVVZUOGA2W5S.json","view_paper":"https://pith.science/paper/2XRCUZV5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2307.01314&json=true","fetch_graph":"https://pith.science/api/pith-number/2XRCUZV5Z7LRFGHVVZUOGA2W5S/graph.json","fetch_events":"https://pith.science/api/pith-number/2XRCUZV5Z7LRFGHVVZUOGA2W5S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2XRCUZV5Z7LRFGHVVZUOGA2W5S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2XRCUZV5Z7LRFGHVVZUOGA2W5S/action/storage_attestation","attest_author":"https://pith.science/pith/2XRCUZV5Z7LRFGHVVZUOGA2W5S/action/author_attestation","sign_citation":"https://pith.science/pith/2XRCUZV5Z7LRFGHVVZUOGA2W5S/action/citation_signature","submit_replication":"https://pith.science/pith/2XRCUZV5Z7LRFGHVVZUOGA2W5S/action/replication_record"}},"created_at":"2026-07-05T06:43:03.708717+00:00","updated_at":"2026-07-05T06:43:03.708717+00:00"}