{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:Y5XJT4E5TMJQW5FHMBINV74GAM","short_pith_number":"pith:Y5XJT4E5","canonical_record":{"source":{"id":"2606.02667","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T11:00:52Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"46ed2005cdb1413e0b3bb9b1a8fed5baf496325e52b27bcae2454e3993840c64","abstract_canon_sha256":"a9ed65920d73e1163cf1d6627acb3ac9a53784f53aeb65647f76771f3610f0ff"},"schema_version":"1.0"},"canonical_sha256":"c76e99f09d9b130b74a76050daff860313afb247d343a168e16db546d8a1b137","source":{"kind":"arxiv","id":"2606.02667","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02667","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02667v1","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02667","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"pith_short_12","alias_value":"Y5XJT4E5TMJQ","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"pith_short_16","alias_value":"Y5XJT4E5TMJQW5FH","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"pith_short_8","alias_value":"Y5XJT4E5","created_at":"2026-06-03T00:05:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:Y5XJT4E5TMJQW5FHMBINV74GAM","target":"record","payload":{"canonical_record":{"source":{"id":"2606.02667","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T11:00:52Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"46ed2005cdb1413e0b3bb9b1a8fed5baf496325e52b27bcae2454e3993840c64","abstract_canon_sha256":"a9ed65920d73e1163cf1d6627acb3ac9a53784f53aeb65647f76771f3610f0ff"},"schema_version":"1.0"},"canonical_sha256":"c76e99f09d9b130b74a76050daff860313afb247d343a168e16db546d8a1b137","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T00:05:05.977356Z","signature_b64":"k4KW19KLAIeuuYBIdnaMjoN9BZ3euQ+9o0KfCKAk1wLNMXGLHqzevvYHCK7otRI+X9s5aapNtVPaHfyW/eP6CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c76e99f09d9b130b74a76050daff860313afb247d343a168e16db546d8a1b137","last_reissued_at":"2026-06-03T00:05:05.976981Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T00:05:05.976981Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.02667","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-03T00:05:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2cQPBxusozpgWtvEbntLmB3evg7fnCd1jftXSNrML67xQKCzv99RSTtklQFMENfoszov+xVk4rRsGd8OL+c6Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T19:28:31.402955Z"},"content_sha256":"6a052750e7a11b1e72e1d154cf675c3d810ef07be4642329b7fde0ba03dd6a70","schema_version":"1.0","event_id":"sha256:6a052750e7a11b1e72e1d154cf675c3d810ef07be4642329b7fde0ba03dd6a70"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:Y5XJT4E5TMJQW5FHMBINV74GAM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Erd\\H{o}s Rado Sunflower (Conjecture) Theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Tapas Kumar Mishra","submitted_at":"2026-06-01T11:00:52Z","abstract_excerpt":"Let $f(k,s)$ denote the minimum integer $m$ such that any family $\\mathcal{F}$ consisting of $k$-sized sets of cardinality at least $m$ always contain a sunflower of size $s$. The Erd\\H{o}s-Rado Sunflower Conjecture states that for every $s >2$, there is an constant $C=C(s)$ such that $f(k,s) \\leq C^k$. In this paper, we prove the conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02667","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.02667/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-03T00:05:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vo7G7RPp9c4LC6z168EuG+c+VYpkm/IgFRhd7Oyqcnx1tROdWUoq6zWG/v5uVrLV8uRIKeDEpaIoGWm8k2dOBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T19:28:31.403363Z"},"content_sha256":"fae09ceb65d61dd55e8b6a3fdde771afe9fbc60572e73613131b5f3da2532168","schema_version":"1.0","event_id":"sha256:fae09ceb65d61dd55e8b6a3fdde771afe9fbc60572e73613131b5f3da2532168"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y5XJT4E5TMJQW5FHMBINV74GAM/bundle.json","state_url":"https://pith.science/pith/Y5XJT4E5TMJQW5FHMBINV74GAM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y5XJT4E5TMJQW5FHMBINV74GAM/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":"2026-06-29T19:28:31Z","links":{"resolver":"https://pith.science/pith/Y5XJT4E5TMJQW5FHMBINV74GAM","bundle":"https://pith.science/pith/Y5XJT4E5TMJQW5FHMBINV74GAM/bundle.json","state":"https://pith.science/pith/Y5XJT4E5TMJQW5FHMBINV74GAM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y5XJT4E5TMJQW5FHMBINV74GAM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:Y5XJT4E5TMJQW5FHMBINV74GAM","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":"a9ed65920d73e1163cf1d6627acb3ac9a53784f53aeb65647f76771f3610f0ff","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T11:00:52Z","title_canon_sha256":"46ed2005cdb1413e0b3bb9b1a8fed5baf496325e52b27bcae2454e3993840c64"},"schema_version":"1.0","source":{"id":"2606.02667","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02667","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02667v1","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02667","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"pith_short_12","alias_value":"Y5XJT4E5TMJQ","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"pith_short_16","alias_value":"Y5XJT4E5TMJQW5FH","created_at":"2026-06-03T00:05:05Z"},{"alias_kind":"pith_short_8","alias_value":"Y5XJT4E5","created_at":"2026-06-03T00:05:05Z"}],"graph_snapshots":[{"event_id":"sha256:fae09ceb65d61dd55e8b6a3fdde771afe9fbc60572e73613131b5f3da2532168","target":"graph","created_at":"2026-06-03T00:05:05Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.02667/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $f(k,s)$ denote the minimum integer $m$ such that any family $\\mathcal{F}$ consisting of $k$-sized sets of cardinality at least $m$ always contain a sunflower of size $s$. The Erd\\H{o}s-Rado Sunflower Conjecture states that for every $s >2$, there is an constant $C=C(s)$ such that $f(k,s) \\leq C^k$. In this paper, we prove the conjecture.","authors_text":"Tapas Kumar Mishra","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T11:00:52Z","title":"Erd\\H{o}s Rado Sunflower (Conjecture) Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02667","kind":"arxiv","version":1},"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":"sha256:6a052750e7a11b1e72e1d154cf675c3d810ef07be4642329b7fde0ba03dd6a70","target":"record","created_at":"2026-06-03T00:05:05Z","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":"a9ed65920d73e1163cf1d6627acb3ac9a53784f53aeb65647f76771f3610f0ff","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T11:00:52Z","title_canon_sha256":"46ed2005cdb1413e0b3bb9b1a8fed5baf496325e52b27bcae2454e3993840c64"},"schema_version":"1.0","source":{"id":"2606.02667","kind":"arxiv","version":1}},"canonical_sha256":"c76e99f09d9b130b74a76050daff860313afb247d343a168e16db546d8a1b137","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c76e99f09d9b130b74a76050daff860313afb247d343a168e16db546d8a1b137","first_computed_at":"2026-06-03T00:05:05.976981Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T00:05:05.976981Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k4KW19KLAIeuuYBIdnaMjoN9BZ3euQ+9o0KfCKAk1wLNMXGLHqzevvYHCK7otRI+X9s5aapNtVPaHfyW/eP6CQ==","signature_status":"signed_v1","signed_at":"2026-06-03T00:05:05.977356Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.02667","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6a052750e7a11b1e72e1d154cf675c3d810ef07be4642329b7fde0ba03dd6a70","sha256:fae09ceb65d61dd55e8b6a3fdde771afe9fbc60572e73613131b5f3da2532168"],"state_sha256":"5ef845c440333a62436572d35fed2c8aab13a0e13abd382f798fae734494dba4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l5D5Nd2xhO1T58w+iyEEVlvr/8sIv7KYjC6kr302wnzi0gfVjSQj0rXfLOCQi6ueJGsucbVCV7YxOR0vUaa0Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T19:28:31.405516Z","bundle_sha256":"635d4e75ef8ad39c0be47e23b060e4829044bfd57019610f1153b1e7cf51bff8"}}