{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:SMDGXC2INVWU7A2V666MAWKXF7","short_pith_number":"pith:SMDGXC2I","canonical_record":{"source":{"id":"1709.06130","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T19:16:34Z","cross_cats_sorted":[],"title_canon_sha256":"78ee974da41f05257ab56f003e44d87930d9cacf1e8524cadc9d1feb11fe20cd","abstract_canon_sha256":"f6234e28992b128c00e8645efa30fbcd843228a799c6783aa943d34d7b307d37"},"schema_version":"1.0"},"canonical_sha256":"93066b8b486d6d4f8355f7bcc059572ff4b0ba289effd584d2314b76585d1785","source":{"kind":"arxiv","id":"1709.06130","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.06130","created_at":"2026-05-18T00:33:58Z"},{"alias_kind":"arxiv_version","alias_value":"1709.06130v2","created_at":"2026-05-18T00:33:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.06130","created_at":"2026-05-18T00:33:58Z"},{"alias_kind":"pith_short_12","alias_value":"SMDGXC2INVWU","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SMDGXC2INVWU7A2V","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SMDGXC2I","created_at":"2026-05-18T12:31:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:SMDGXC2INVWU7A2V666MAWKXF7","target":"record","payload":{"canonical_record":{"source":{"id":"1709.06130","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T19:16:34Z","cross_cats_sorted":[],"title_canon_sha256":"78ee974da41f05257ab56f003e44d87930d9cacf1e8524cadc9d1feb11fe20cd","abstract_canon_sha256":"f6234e28992b128c00e8645efa30fbcd843228a799c6783aa943d34d7b307d37"},"schema_version":"1.0"},"canonical_sha256":"93066b8b486d6d4f8355f7bcc059572ff4b0ba289effd584d2314b76585d1785","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:58.399756Z","signature_b64":"V+ib+qGoqKcPQW7DCkap9/fPXAJ/91beNs0LUv2Rs0rUOa9XWmUF6kmKADfBkXNoyxRpLO5gtYGHmO4++FJqAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93066b8b486d6d4f8355f7bcc059572ff4b0ba289effd584d2314b76585d1785","last_reissued_at":"2026-05-18T00:33:58.399154Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:58.399154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.06130","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:33:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VYPr7r863yjsr/U0iytO9uv67Xws9dNa5Ntz5QslslEG5BTv652HT2YZiALS6F84WK4ydX0H3mhHafFF/TikAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T15:55:16.166405Z"},"content_sha256":"bf79dca08b309f2c423403f84375eed0fa5411be595005c1510201a507539041","schema_version":"1.0","event_id":"sha256:bf79dca08b309f2c423403f84375eed0fa5411be595005c1510201a507539041"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:SMDGXC2INVWU7A2V666MAWKXF7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gallai-Ramsey numbers of $C_9$ with multiple colors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Bosse, Zi-Xia Song","submitted_at":"2017-09-18T19:16:34Z","abstract_excerpt":"We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $G$. It turns out that $gr_k(K_3, G)$ behaves more nicely than the classical Ramsey number $r_k(G)$. However, finding exact values of $gr_k (K_3, G)$ is far from trivial. In this paper, we prove that $gr_k(K_3, C_9)= 4\\cdot 2^k+1$ for all $k\\ge1$. This new result provides partial evidence fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.06130","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:33:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rytxj6ZIZDPemToP3Yvsmf62xm2OwbFGjPY/VLiNTEpt4J4o4yvfYIcuyHXuBGzqT0CcW3BYBikA2HzUzpfxBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T15:55:16.166792Z"},"content_sha256":"ea70d7f9bf61cc00d2534afeef805be6a00afe531551666a9f8af007f24127ff","schema_version":"1.0","event_id":"sha256:ea70d7f9bf61cc00d2534afeef805be6a00afe531551666a9f8af007f24127ff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SMDGXC2INVWU7A2V666MAWKXF7/bundle.json","state_url":"https://pith.science/pith/SMDGXC2INVWU7A2V666MAWKXF7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SMDGXC2INVWU7A2V666MAWKXF7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T15:55:16Z","links":{"resolver":"https://pith.science/pith/SMDGXC2INVWU7A2V666MAWKXF7","bundle":"https://pith.science/pith/SMDGXC2INVWU7A2V666MAWKXF7/bundle.json","state":"https://pith.science/pith/SMDGXC2INVWU7A2V666MAWKXF7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SMDGXC2INVWU7A2V666MAWKXF7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:SMDGXC2INVWU7A2V666MAWKXF7","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":"f6234e28992b128c00e8645efa30fbcd843228a799c6783aa943d34d7b307d37","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T19:16:34Z","title_canon_sha256":"78ee974da41f05257ab56f003e44d87930d9cacf1e8524cadc9d1feb11fe20cd"},"schema_version":"1.0","source":{"id":"1709.06130","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.06130","created_at":"2026-05-18T00:33:58Z"},{"alias_kind":"arxiv_version","alias_value":"1709.06130v2","created_at":"2026-05-18T00:33:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.06130","created_at":"2026-05-18T00:33:58Z"},{"alias_kind":"pith_short_12","alias_value":"SMDGXC2INVWU","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SMDGXC2INVWU7A2V","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SMDGXC2I","created_at":"2026-05-18T12:31:43Z"}],"graph_snapshots":[{"event_id":"sha256:ea70d7f9bf61cc00d2534afeef805be6a00afe531551666a9f8af007f24127ff","target":"graph","created_at":"2026-05-18T00:33:58Z","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":"We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $G$. It turns out that $gr_k(K_3, G)$ behaves more nicely than the classical Ramsey number $r_k(G)$. However, finding exact values of $gr_k (K_3, G)$ is far from trivial. In this paper, we prove that $gr_k(K_3, C_9)= 4\\cdot 2^k+1$ for all $k\\ge1$. This new result provides partial evidence fo","authors_text":"Christian Bosse, Zi-Xia Song","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T19:16:34Z","title":"Gallai-Ramsey numbers of $C_9$ with multiple colors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.06130","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:bf79dca08b309f2c423403f84375eed0fa5411be595005c1510201a507539041","target":"record","created_at":"2026-05-18T00:33:58Z","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":"f6234e28992b128c00e8645efa30fbcd843228a799c6783aa943d34d7b307d37","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T19:16:34Z","title_canon_sha256":"78ee974da41f05257ab56f003e44d87930d9cacf1e8524cadc9d1feb11fe20cd"},"schema_version":"1.0","source":{"id":"1709.06130","kind":"arxiv","version":2}},"canonical_sha256":"93066b8b486d6d4f8355f7bcc059572ff4b0ba289effd584d2314b76585d1785","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"93066b8b486d6d4f8355f7bcc059572ff4b0ba289effd584d2314b76585d1785","first_computed_at":"2026-05-18T00:33:58.399154Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:58.399154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"V+ib+qGoqKcPQW7DCkap9/fPXAJ/91beNs0LUv2Rs0rUOa9XWmUF6kmKADfBkXNoyxRpLO5gtYGHmO4++FJqAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:58.399756Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.06130","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf79dca08b309f2c423403f84375eed0fa5411be595005c1510201a507539041","sha256:ea70d7f9bf61cc00d2534afeef805be6a00afe531551666a9f8af007f24127ff"],"state_sha256":"943696ba56941e9915667cde1fdf16e10d7ea922e4546932c8d1e4fc5916d4b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5tSCl7ZbfFhsQgfle4k4Dl5kkZqYl56eNszXbMzLgDAfbw5RRR6NjlkXQTtaYBYFDR2IFtWW/+kji7YKZu03AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T15:55:16.168813Z","bundle_sha256":"beacb9dd18123ad755f6f982823fde2ec612d3e2dfaae87d4af0799d3ff627c0"}}