{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:57CT3KQ2EUQ6B74P5Y7HFLCGW5","short_pith_number":"pith:57CT3KQ2","canonical_record":{"source":{"id":"1709.06163","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T20:43:44Z","cross_cats_sorted":[],"title_canon_sha256":"7c0478face0200180ab0cb9dd8e4a85fe14f2de31662414589adec7e22ba6644","abstract_canon_sha256":"eaf55ca0f340dfd636ea60804d07c508d7bb812e4ef780a4d7cdea8f7c05284a"},"schema_version":"1.0"},"canonical_sha256":"efc53daa1a2521e0ff8fee3e72ac46b7780e57371a153e3c5130a145198dbc4a","source":{"kind":"arxiv","id":"1709.06163","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.06163","created_at":"2026-05-17T23:43:54Z"},{"alias_kind":"arxiv_version","alias_value":"1709.06163v2","created_at":"2026-05-17T23:43:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.06163","created_at":"2026-05-17T23:43:54Z"},{"alias_kind":"pith_short_12","alias_value":"57CT3KQ2EUQ6","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"57CT3KQ2EUQ6B74P","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"57CT3KQ2","created_at":"2026-05-18T12:31:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:57CT3KQ2EUQ6B74P5Y7HFLCGW5","target":"record","payload":{"canonical_record":{"source":{"id":"1709.06163","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T20:43:44Z","cross_cats_sorted":[],"title_canon_sha256":"7c0478face0200180ab0cb9dd8e4a85fe14f2de31662414589adec7e22ba6644","abstract_canon_sha256":"eaf55ca0f340dfd636ea60804d07c508d7bb812e4ef780a4d7cdea8f7c05284a"},"schema_version":"1.0"},"canonical_sha256":"efc53daa1a2521e0ff8fee3e72ac46b7780e57371a153e3c5130a145198dbc4a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:54.069915Z","signature_b64":"4aYbkCwvQISgot9bEGs5roUC/0P2xFli0+BIBNg06iZTt+mLxVljAmc1weGkqiAFNWmRAx2gAkvemR2tG6/tCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"efc53daa1a2521e0ff8fee3e72ac46b7780e57371a153e3c5130a145198dbc4a","last_reissued_at":"2026-05-17T23:43:54.069221Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:54.069221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.06163","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":"2026-05-17T23:43:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jF9AQwT72/fTfyX7J4x1BAlrB86O88y+LuKotbvfIo9G+mVNkR9r8YbkaWcIMxkSECL0P1XSvE0105dn0EmMAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T20:16:49.924756Z"},"content_sha256":"f5e3fe7946999f65129a0c5114b435e5a3169d8f253d5f288c5cbc69209dcb26","schema_version":"1.0","event_id":"sha256:f5e3fe7946999f65129a0c5114b435e5a3169d8f253d5f288c5cbc69209dcb26"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:57CT3KQ2EUQ6B74P5Y7HFLCGW5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Many Triangles with Few Edges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. J. Radcliffe, R. Kirsch","submitted_at":"2017-09-18T20:43:44Z","abstract_excerpt":"Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with $n$ vertices and maximum degree at most $r$, where $n = a(r+1)+b$ and $0 \\le b \\le r$, $aK_{r+1}\\cup K_b$ has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh, and Sudakov conjectured that $aK_{r+1}\\cup K_b$ also maximizes the number of complete subgraphs $K_t$ for each fixed size $t \\ge 3$, and proved this for $a = 1$. Cutler and Radcliffe proved this conjecture for $r \\le 6$.\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.06163","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":"2026-05-17T23:43:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jqXwT2ixVZhOJOEcelo66+x+7csDd/PknP6pNZfOyX4o9RUDArFgYmdKLE5LhKmpS/Xn2+2H7WRidVeVP66ODw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T20:16:49.925094Z"},"content_sha256":"22632f6e7f73e1a1a21184bcddb2f749e6173d18a8a5e1d3739b663c5673c528","schema_version":"1.0","event_id":"sha256:22632f6e7f73e1a1a21184bcddb2f749e6173d18a8a5e1d3739b663c5673c528"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/57CT3KQ2EUQ6B74P5Y7HFLCGW5/bundle.json","state_url":"https://pith.science/pith/57CT3KQ2EUQ6B74P5Y7HFLCGW5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/57CT3KQ2EUQ6B74P5Y7HFLCGW5/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":"2026-06-01T20:16:49Z","links":{"resolver":"https://pith.science/pith/57CT3KQ2EUQ6B74P5Y7HFLCGW5","bundle":"https://pith.science/pith/57CT3KQ2EUQ6B74P5Y7HFLCGW5/bundle.json","state":"https://pith.science/pith/57CT3KQ2EUQ6B74P5Y7HFLCGW5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/57CT3KQ2EUQ6B74P5Y7HFLCGW5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:57CT3KQ2EUQ6B74P5Y7HFLCGW5","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":"eaf55ca0f340dfd636ea60804d07c508d7bb812e4ef780a4d7cdea8f7c05284a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T20:43:44Z","title_canon_sha256":"7c0478face0200180ab0cb9dd8e4a85fe14f2de31662414589adec7e22ba6644"},"schema_version":"1.0","source":{"id":"1709.06163","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.06163","created_at":"2026-05-17T23:43:54Z"},{"alias_kind":"arxiv_version","alias_value":"1709.06163v2","created_at":"2026-05-17T23:43:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.06163","created_at":"2026-05-17T23:43:54Z"},{"alias_kind":"pith_short_12","alias_value":"57CT3KQ2EUQ6","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"57CT3KQ2EUQ6B74P","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"57CT3KQ2","created_at":"2026-05-18T12:31:00Z"}],"graph_snapshots":[{"event_id":"sha256:22632f6e7f73e1a1a21184bcddb2f749e6173d18a8a5e1d3739b663c5673c528","target":"graph","created_at":"2026-05-17T23:43:54Z","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":"Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with $n$ vertices and maximum degree at most $r$, where $n = a(r+1)+b$ and $0 \\le b \\le r$, $aK_{r+1}\\cup K_b$ has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh, and Sudakov conjectured that $aK_{r+1}\\cup K_b$ also maximizes the number of complete subgraphs $K_t$ for each fixed size $t \\ge 3$, and proved this for $a = 1$. Cutler and Radcliffe proved this conjecture for $r \\le 6$.\n ","authors_text":"A. J. Radcliffe, R. Kirsch","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T20:43:44Z","title":"Many Triangles with Few Edges"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.06163","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:f5e3fe7946999f65129a0c5114b435e5a3169d8f253d5f288c5cbc69209dcb26","target":"record","created_at":"2026-05-17T23:43:54Z","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":"eaf55ca0f340dfd636ea60804d07c508d7bb812e4ef780a4d7cdea8f7c05284a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T20:43:44Z","title_canon_sha256":"7c0478face0200180ab0cb9dd8e4a85fe14f2de31662414589adec7e22ba6644"},"schema_version":"1.0","source":{"id":"1709.06163","kind":"arxiv","version":2}},"canonical_sha256":"efc53daa1a2521e0ff8fee3e72ac46b7780e57371a153e3c5130a145198dbc4a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"efc53daa1a2521e0ff8fee3e72ac46b7780e57371a153e3c5130a145198dbc4a","first_computed_at":"2026-05-17T23:43:54.069221Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:54.069221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4aYbkCwvQISgot9bEGs5roUC/0P2xFli0+BIBNg06iZTt+mLxVljAmc1weGkqiAFNWmRAx2gAkvemR2tG6/tCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:54.069915Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.06163","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f5e3fe7946999f65129a0c5114b435e5a3169d8f253d5f288c5cbc69209dcb26","sha256:22632f6e7f73e1a1a21184bcddb2f749e6173d18a8a5e1d3739b663c5673c528"],"state_sha256":"d20fd95fa0a057f17807883d532b72999e9a07e19d79813a9e77a663c1689cb0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TgOsQcmTlTO63WklfAQUxbg51OcMAWlptkzlb0ul9hBqVsz6m50pBPd+bRAeqGRatDLe1kz26qb7rGpcDCeoBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T20:16:49.927131Z","bundle_sha256":"fc76e7cef9cd4cad17593b2f5b56e5c62bb237175d5d6408891c25cda4715fc4"}}