{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:OY6GZGWHNSM73X4SXWZUYYJIUU","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":"d3b02224e270843b97d2e64df2c6a47b2d3825cfa962ad000eee716950861f08","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-14T16:47:30Z","title_canon_sha256":"60ce4a0f35770eaf48e339e6f24f1a3adeaba10190c48f7c750157e759ce0771"},"schema_version":"1.0","source":{"id":"1601.03659","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03659","created_at":"2026-05-18T01:22:52Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03659v1","created_at":"2026-05-18T01:22:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03659","created_at":"2026-05-18T01:22:52Z"},{"alias_kind":"pith_short_12","alias_value":"OY6GZGWHNSM7","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OY6GZGWHNSM73X4S","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OY6GZGWH","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:68ab9b2b772cead88c29bc42f72dde647a13e77454b537e7b4a6a2ad0fb4fc0a","target":"graph","created_at":"2026-05-18T01:22:52Z","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":"A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most $(1+o(1))\\binom{n}{n/2}$. Later, Burosch--Demetrovics-Katona-Kleitman-Sapozhenko asked for the number $\\alpha(n)$ of such families, and they proved that $2^{\\binom{n}{n/2}}\\leq \\alpha(n) \\leq 2^{2\\sqrt{2}\\binom{n}{n/2}(1+o(1))}$. They conjectured that the constant $2\\sqrt{2}$ can be removed in the exponent of the right hand side. We prove their conjecture by formulating a new container-ty","authors_text":"Adam Zsolt Wagner, Jozsef Balogh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-14T16:47:30Z","title":"On the number of union-free families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03659","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":"sha256:1a5bf81ea47848ad176971786e993a78547393205040ca9455c90bd31f1e47b3","target":"record","created_at":"2026-05-18T01:22:52Z","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":"d3b02224e270843b97d2e64df2c6a47b2d3825cfa962ad000eee716950861f08","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-14T16:47:30Z","title_canon_sha256":"60ce4a0f35770eaf48e339e6f24f1a3adeaba10190c48f7c750157e759ce0771"},"schema_version":"1.0","source":{"id":"1601.03659","kind":"arxiv","version":1}},"canonical_sha256":"763c6c9ac76c99fddf92bdb34c6128a510e0c70063741ed5f9d67c9625ac36af","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"763c6c9ac76c99fddf92bdb34c6128a510e0c70063741ed5f9d67c9625ac36af","first_computed_at":"2026-05-18T01:22:52.911260Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:52.911260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YJVcUY2Z4MXJYAPhxevkhAW6/l+8Ss5XkD3noQGktAU9tzYL5cQVwD0T4IPGx8Wr5WItgli4PY2FwNziObRlDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:52.911786Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.03659","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1a5bf81ea47848ad176971786e993a78547393205040ca9455c90bd31f1e47b3","sha256:68ab9b2b772cead88c29bc42f72dde647a13e77454b537e7b4a6a2ad0fb4fc0a"],"state_sha256":"71be13ae4605f47d2a45fb32d53acfbab54cad69da57fe4639a15d515d3146c0"}