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A $t$-edge colored graph $G$ is $(\\mathcal{C}, t)$-saturated if $G$ does not contain any graph in $\\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some graph in $\\mathcal{C}$. Similarly to classical saturation functions, define $\\mathrm{sat}_t(n, \\mathcal{C})$ to be the minimum number of edges in a $(\\mathcal{C},t)$ saturated graph. Let $\\mathcal{C}_r(H)$ be the family consisting of every edge-colored copy of $H$ which uses exactly $r$ colors.\n  In this paper we consider a variety of colored saturation prob"},"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":"1712.00163","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-01T02:28:29Z","cross_cats_sorted":[],"title_canon_sha256":"c21dd79304ddf5452acaa5fcc0e5db7997cc0110a9f21ae09275fd9242c0258b","abstract_canon_sha256":"4b60d863f06cfe0adb3abc62433ed37d552b6391c0525f81102917fda28bba88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:07.121857Z","signature_b64":"zIHeANimPxB8sZ6K2O0UpNXCdt2Bm4KUHSilc0yGHCiNxdZycMSLNIhgR3LW7+gRzPptI4uFMTUdw6vBvG2PCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53d56a85573c9ebece3de6832414c89a5d35af7ac389912ee4acab5a30b1b58d","last_reissued_at":"2026-05-18T00:29:07.121196Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:07.121196Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Edge-Colored Saturation Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Schulte, Casey Tompkins, Daniel Johnston, Eric Sullivan, Florian Pfender, Heather C. 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Let $\\mathcal{C}_r(H)$ be the family consisting of every edge-colored copy of $H$ which uses exactly $r$ colors.\n  In this paper we consider a variety of colored saturation prob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.00163","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":""},"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":"1712.00163","created_at":"2026-05-18T00:29:07.121296+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.00163v1","created_at":"2026-05-18T00:29:07.121296+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.00163","created_at":"2026-05-18T00:29:07.121296+00:00"},{"alias_kind":"pith_short_12","alias_value":"KPKWVBKXHSPL","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_16","alias_value":"KPKWVBKXHSPL5TR5","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_8","alias_value":"KPKWVBKX","created_at":"2026-05-18T12:31:24.725408+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KPKWVBKXHSPL5TR542BSIFGITJ","json":"https://pith.science/pith/KPKWVBKXHSPL5TR542BSIFGITJ.json","graph_json":"https://pith.science/api/pith-number/KPKWVBKXHSPL5TR542BSIFGITJ/graph.json","events_json":"https://pith.science/api/pith-number/KPKWVBKXHSPL5TR542BSIFGITJ/events.json","paper":"https://pith.science/paper/KPKWVBKX"},"agent_actions":{"view_html":"https://pith.science/pith/KPKWVBKXHSPL5TR542BSIFGITJ","download_json":"https://pith.science/pith/KPKWVBKXHSPL5TR542BSIFGITJ.json","view_paper":"https://pith.science/paper/KPKWVBKX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.00163&json=true","fetch_graph":"https://pith.science/api/pith-number/KPKWVBKXHSPL5TR542BSIFGITJ/graph.json","fetch_events":"https://pith.science/api/pith-number/KPKWVBKXHSPL5TR542BSIFGITJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KPKWVBKXHSPL5TR542BSIFGITJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KPKWVBKXHSPL5TR542BSIFGITJ/action/storage_attestation","attest_author":"https://pith.science/pith/KPKWVBKXHSPL5TR542BSIFGITJ/action/author_attestation","sign_citation":"https://pith.science/pith/KPKWVBKXHSPL5TR542BSIFGITJ/action/citation_signature","submit_replication":"https://pith.science/pith/KPKWVBKXHSPL5TR542BSIFGITJ/action/replication_record"}},"created_at":"2026-05-18T00:29:07.121296+00:00","updated_at":"2026-05-18T00:29:07.121296+00:00"}