{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:YNJH2ZN2EVPOXWWL7PBAOJIY4X","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6e795221f9f1a65655578504ef0612fd6b74a7486328bc8062ced927d4eb683c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-26T18:59:12Z","title_canon_sha256":"cf2765b53a3142a8d598cfcec703ce447ead27b960777cb9bd67ae2770d82e4a"},"schema_version":"1.0","source":{"id":"1812.10465","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.10465","created_at":"2026-05-17T23:57:18Z"},{"alias_kind":"arxiv_version","alias_value":"1812.10465v2","created_at":"2026-05-17T23:57:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.10465","created_at":"2026-05-17T23:57:18Z"},{"alias_kind":"pith_short_12","alias_value":"YNJH2ZN2EVPO","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"YNJH2ZN2EVPOXWWL","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"YNJH2ZN2","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:298a236ea4fa4afe6fa7b6dc2241631c9d90b5532780fd7decb8ea72a3001bb9","target":"graph","created_at":"2026-05-17T23:57:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$ with $k\\le 2^{n/4300}$, the number of Gallai colorings of $K_n$ that use at most $k$ given colors is $(\\binom{k}{2}+o_n(1))\\,2^{\\binom{n}{2}}$. Our result is asymptotically best possible and implies that, for those $k$, almost all Gallai $k$-colorings use only two colors. However, this is not true for $k \\ge \\Omega (2^{2n})$.","authors_text":"Fabricio S. Benevides, Jie Han, Josefran de Oliveira Bastos","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-26T18:59:12Z","title":"The number of Gallai k-colorings of complete graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.10465","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1f5759e040a40877f732651e9dfc32a8bb86b1fe44e704524cc378dd83ca7c5f","target":"record","created_at":"2026-05-17T23:57:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6e795221f9f1a65655578504ef0612fd6b74a7486328bc8062ced927d4eb683c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-26T18:59:12Z","title_canon_sha256":"cf2765b53a3142a8d598cfcec703ce447ead27b960777cb9bd67ae2770d82e4a"},"schema_version":"1.0","source":{"id":"1812.10465","kind":"arxiv","version":2}},"canonical_sha256":"c3527d65ba255eebdacbfbc2072518e5f99ec06a7f7002eda09bf64ecd4827ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c3527d65ba255eebdacbfbc2072518e5f99ec06a7f7002eda09bf64ecd4827ae","first_computed_at":"2026-05-17T23:57:18.961376Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:18.961376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ymp5Oh8Ta93TQizgPINi0m17xnN8D/Oo7InkaemmKyXAIl8rPk7NK3qyX9ieEyCOC9kIfpo+s1LH2rROuSIyDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:18.962056Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.10465","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1f5759e040a40877f732651e9dfc32a8bb86b1fe44e704524cc378dd83ca7c5f","sha256:298a236ea4fa4afe6fa7b6dc2241631c9d90b5532780fd7decb8ea72a3001bb9"],"state_sha256":"a35b516850e5456ba6ca92de43ae8d066333a29ed8245db15595deb9824db464"}