{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:ZZCSLL2V6CUXPNUD2VCSDHSHX3","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":"ddf51788afeae67cb3bab08955f08cf20e3a0fa878bff3cba7fb8f56b6f151f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-30T15:12:33Z","title_canon_sha256":"31da352b3c49bdf1da75df9ac6216cae81b0f39afa9972d7e55f3b41b55a7fb5"},"schema_version":"1.0","source":{"id":"1804.11269","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.11269","created_at":"2026-05-18T00:17:14Z"},{"alias_kind":"arxiv_version","alias_value":"1804.11269v1","created_at":"2026-05-18T00:17:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.11269","created_at":"2026-05-18T00:17:14Z"},{"alias_kind":"pith_short_12","alias_value":"ZZCSLL2V6CUX","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"ZZCSLL2V6CUXPNUD","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"ZZCSLL2V","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:5ee26eac53bd02f8e3267e3d81a79148d5ed4cee58272beb1c06fc6e49d07f83","target":"graph","created_at":"2026-05-18T00:17:14Z","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":"In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families $\\mathcal{A}, \\mathcal{B} \\subset \\binom{[n]}{k}$, they must satisfy $\\min\\{|\\mathcal{A}|, |\\mathcal{B}|\\} \\le \\frac{1}{2} \\binom{n-1}{k-1}$? We give an affirmative answer for $n \\ge 2k^2$, and construct families showing that this range is essentially the best one could hope for, up to a constant factor. The second problem is a conjecture of Frankl. It states that for $n \\ge 3k$, th","authors_text":"Hao Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-30T15:12:33Z","title":"Two extremal problems on intersecting families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.11269","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":"sha256:402074ad33248d1f0a53b4ed0680cc814c7d2ea5a4d000d4a1bcaf12a9b7df40","target":"record","created_at":"2026-05-18T00:17:14Z","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":"ddf51788afeae67cb3bab08955f08cf20e3a0fa878bff3cba7fb8f56b6f151f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-30T15:12:33Z","title_canon_sha256":"31da352b3c49bdf1da75df9ac6216cae81b0f39afa9972d7e55f3b41b55a7fb5"},"schema_version":"1.0","source":{"id":"1804.11269","kind":"arxiv","version":1}},"canonical_sha256":"ce4525af55f0a977b683d545219e47becde985bd3e13884f625f63e9c74f7bb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce4525af55f0a977b683d545219e47becde985bd3e13884f625f63e9c74f7bb4","first_computed_at":"2026-05-18T00:17:14.073594Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:14.073594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7L5REnOkb6Y8XvLWAeCdpfQMVXxFIMwZB10ePWHYp/CMPOqk/DMOrWZ0HMoBOd6pfP9OGAX/kI3PkRUWttL9Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:14.074187Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.11269","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:402074ad33248d1f0a53b4ed0680cc814c7d2ea5a4d000d4a1bcaf12a9b7df40","sha256:5ee26eac53bd02f8e3267e3d81a79148d5ed4cee58272beb1c06fc6e49d07f83"],"state_sha256":"0201eeeeaa26ccc3b97a0d820398e473794cc53a547dfdb93bdd8c276f2b6f5e"}