{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:CFY2IKIQJV2KFM33Q3QPTC4JTN","short_pith_number":"pith:CFY2IKIQ","canonical_record":{"source":{"id":"2211.09055","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-11-16T17:17:42Z","cross_cats_sorted":[],"title_canon_sha256":"37cfec0a080cd456b20d1fff20238aca3d0d097f74dd1a929c0348c649248c76","abstract_canon_sha256":"d9917eec784b192a22e6016b833e152494d85c602e19cb73c7540c6671b21c37"},"schema_version":"1.0"},"canonical_sha256":"1171a429104d74a2b37b86e0f98b899b7ddcc78233c19efbb1ce6b434da94874","source":{"kind":"arxiv","id":"2211.09055","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2211.09055","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"arxiv_version","alias_value":"2211.09055v2","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.09055","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"pith_short_12","alias_value":"CFY2IKIQJV2K","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"pith_short_16","alias_value":"CFY2IKIQJV2KFM33","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"pith_short_8","alias_value":"CFY2IKIQ","created_at":"2026-07-05T05:19:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:CFY2IKIQJV2KFM33Q3QPTC4JTN","target":"record","payload":{"canonical_record":{"source":{"id":"2211.09055","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-11-16T17:17:42Z","cross_cats_sorted":[],"title_canon_sha256":"37cfec0a080cd456b20d1fff20238aca3d0d097f74dd1a929c0348c649248c76","abstract_canon_sha256":"d9917eec784b192a22e6016b833e152494d85c602e19cb73c7540c6671b21c37"},"schema_version":"1.0"},"canonical_sha256":"1171a429104d74a2b37b86e0f98b899b7ddcc78233c19efbb1ce6b434da94874","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:19:56.118307Z","signature_b64":"F7TKiwjs6wgnr8gMHwGL+ENFDxLR+NDB56s2k+ExGb6dOc6WC/vZGrGupwR0WkygdY//9vo3aPBXx4bwYfnBBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1171a429104d74a2b37b86e0f98b899b7ddcc78233c19efbb1ce6b434da94874","last_reissued_at":"2026-07-05T05:19:56.117876Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:19:56.117876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2211.09055","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":"2026-07-05T05:19:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BJry3V/hR0n/s0NnV1DPigIQFdFP18rlD+8yQTXfR0GalMiG42SFP9fjJljB+JQRMhn6lzTU4muSaVW0rwpLDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T07:51:40.645760Z"},"content_sha256":"900a4455c69896314eb3f61459093fe92690eacf419a2bd19fb32bc13d98e4eb","schema_version":"1.0","event_id":"sha256:900a4455c69896314eb3f61459093fe92690eacf419a2bd19fb32bc13d98e4eb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:CFY2IKIQJV2KFM33Q3QPTC4JTN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A constant lower bound for the union-closed sets conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Justin Gilmer","submitted_at":"2022-11-16T17:17:42Z","abstract_excerpt":"We show that for any union-closed family $\\mathcal{F} \\subseteq 2^{[n]}, \\mathcal{F} \\neq \\{\\emptyset\\}$, there exists an $i \\in [n]$ which is contained in a $0.01$ fraction of the sets in $\\mathcal{F}$. This is the first known constant lower bound, and improves upon the $\\Omega(\\log_2(|\\mathcal{F}|)^{-1})$ bounds of Knill and W\\'{o}jick. Our result follows from an information theoretic strengthening of the conjecture. Specifically, we show that if $A, B$ are independent samples from a distribution over subsets of $[n]$ such that $Pr[i \\in A] < 0.01$ for all $i$ and $H(A) > 0$, then $H(A \\cup "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.09055","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.09055/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":"2026-07-05T05:19:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tpzHBhuhakxRH8vLN78Un/uYjJGdP1rP5/2LU2NR4GU2yiNFoMOOfO2htc9uKdxn03VHIKC4mIW+qxE8/GqeAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T07:51:40.646266Z"},"content_sha256":"e1ee245ccb4e7e1657ffe4e46721ed6f5d646a29f09e4e47dccfe74af0cfb740","schema_version":"1.0","event_id":"sha256:e1ee245ccb4e7e1657ffe4e46721ed6f5d646a29f09e4e47dccfe74af0cfb740"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CFY2IKIQJV2KFM33Q3QPTC4JTN/bundle.json","state_url":"https://pith.science/pith/CFY2IKIQJV2KFM33Q3QPTC4JTN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CFY2IKIQJV2KFM33Q3QPTC4JTN/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":"2026-08-06T07:51:40Z","links":{"resolver":"https://pith.science/pith/CFY2IKIQJV2KFM33Q3QPTC4JTN","bundle":"https://pith.science/pith/CFY2IKIQJV2KFM33Q3QPTC4JTN/bundle.json","state":"https://pith.science/pith/CFY2IKIQJV2KFM33Q3QPTC4JTN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CFY2IKIQJV2KFM33Q3QPTC4JTN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:CFY2IKIQJV2KFM33Q3QPTC4JTN","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":"d9917eec784b192a22e6016b833e152494d85c602e19cb73c7540c6671b21c37","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-11-16T17:17:42Z","title_canon_sha256":"37cfec0a080cd456b20d1fff20238aca3d0d097f74dd1a929c0348c649248c76"},"schema_version":"1.0","source":{"id":"2211.09055","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2211.09055","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"arxiv_version","alias_value":"2211.09055v2","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.09055","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"pith_short_12","alias_value":"CFY2IKIQJV2K","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"pith_short_16","alias_value":"CFY2IKIQJV2KFM33","created_at":"2026-07-05T05:19:56Z"},{"alias_kind":"pith_short_8","alias_value":"CFY2IKIQ","created_at":"2026-07-05T05:19:56Z"}],"graph_snapshots":[{"event_id":"sha256:e1ee245ccb4e7e1657ffe4e46721ed6f5d646a29f09e4e47dccfe74af0cfb740","target":"graph","created_at":"2026-07-05T05:19:56Z","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/2211.09055/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that for any union-closed family $\\mathcal{F} \\subseteq 2^{[n]}, \\mathcal{F} \\neq \\{\\emptyset\\}$, there exists an $i \\in [n]$ which is contained in a $0.01$ fraction of the sets in $\\mathcal{F}$. This is the first known constant lower bound, and improves upon the $\\Omega(\\log_2(|\\mathcal{F}|)^{-1})$ bounds of Knill and W\\'{o}jick. Our result follows from an information theoretic strengthening of the conjecture. Specifically, we show that if $A, B$ are independent samples from a distribution over subsets of $[n]$ such that $Pr[i \\in A] < 0.01$ for all $i$ and $H(A) > 0$, then $H(A \\cup ","authors_text":"Justin Gilmer","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-11-16T17:17:42Z","title":"A constant lower bound for the union-closed sets conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.09055","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":"sha256:900a4455c69896314eb3f61459093fe92690eacf419a2bd19fb32bc13d98e4eb","target":"record","created_at":"2026-07-05T05:19:56Z","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":"d9917eec784b192a22e6016b833e152494d85c602e19cb73c7540c6671b21c37","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-11-16T17:17:42Z","title_canon_sha256":"37cfec0a080cd456b20d1fff20238aca3d0d097f74dd1a929c0348c649248c76"},"schema_version":"1.0","source":{"id":"2211.09055","kind":"arxiv","version":2}},"canonical_sha256":"1171a429104d74a2b37b86e0f98b899b7ddcc78233c19efbb1ce6b434da94874","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1171a429104d74a2b37b86e0f98b899b7ddcc78233c19efbb1ce6b434da94874","first_computed_at":"2026-07-05T05:19:56.117876Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:19:56.117876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F7TKiwjs6wgnr8gMHwGL+ENFDxLR+NDB56s2k+ExGb6dOc6WC/vZGrGupwR0WkygdY//9vo3aPBXx4bwYfnBBw==","signature_status":"signed_v1","signed_at":"2026-07-05T05:19:56.118307Z","signed_message":"canonical_sha256_bytes"},"source_id":"2211.09055","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:900a4455c69896314eb3f61459093fe92690eacf419a2bd19fb32bc13d98e4eb","sha256:e1ee245ccb4e7e1657ffe4e46721ed6f5d646a29f09e4e47dccfe74af0cfb740"],"state_sha256":"04b7f646913bbcf1ce2ed83e9bb2e5f4e168d814de8cb14f064c0f6a5e0e70ab"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RUdJ60jGlJfBpwly/4Uxiq0lZtke/6eJb4lt2XYdgtQTXiRUvh5s5tbuI+k75wIrWX3ep3Kx1HJq1D0c5ry/DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T07:51:40.650070Z","bundle_sha256":"48098e41031770a7494d19e90a7809adb6cb3a4dce736d43c87ba98c26f6b440"}}