{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:JCST764TVS4OVEZ4NNW33HPFAB","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":"42ed563af623d57948fd756451470da1c24b3c2a6c53ad66f876cf11f48900c5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-22T13:33:17Z","title_canon_sha256":"213d48ca9e91466d8a64541c1eaf0947af845ff4fd80089e454562f34744145a"},"schema_version":"1.0","source":{"id":"2606.23322","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.23322","created_at":"2026-06-23T03:14:16Z"},{"alias_kind":"arxiv_version","alias_value":"2606.23322v1","created_at":"2026-06-23T03:14:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.23322","created_at":"2026-06-23T03:14:16Z"},{"alias_kind":"pith_short_12","alias_value":"JCST764TVS4O","created_at":"2026-06-23T03:14:16Z"},{"alias_kind":"pith_short_16","alias_value":"JCST764TVS4OVEZ4","created_at":"2026-06-23T03:14:16Z"},{"alias_kind":"pith_short_8","alias_value":"JCST764T","created_at":"2026-06-23T03:14:16Z"}],"graph_snapshots":[{"event_id":"sha256:2b387719c6cc950def281f8947e230158c5cbbfcbf03ec73311e7daae9b8f52c","target":"graph","created_at":"2026-06-23T03:14:16Z","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/2606.23322/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Two families $\\mathcal{A}, \\mathcal{B} \\subset \\binom{[n]}{k}$ are cross-intersecting if $A \\cap B \\ne \\emptyset$ for all $A \\in \\mathcal{A}$ and $B \\in \\mathcal{B}$, and non-trivial if neither $\\m A$ nor $\\m B$ is a star. Pyber proved that any two cross-intersecting families $\\mathcal{A}, \\mathcal{B} \\subset \\binom{[n]}{k}$ satisfy $|\\mathcal{A}||\\mathcal{B}| \\le \\binom{n-1}{k-1}^2$, and the maximum is attained by two full stars. Frankl, as well as Frankl and Wang, conjectured that the sharp bound, when both families are required to be non-trivial, is $h(n,k)^2$, where $h(n,k) = \\binom{n-1}{k","authors_text":"Yang Huang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-22T13:33:17Z","title":"A sharp product bound for non-trivial cross-intersecting families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23322","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:e148bd24ae0b9ee9ef197c77e2d963b997cdd6e562c2b13d4410f0429773f068","target":"record","created_at":"2026-06-23T03:14:16Z","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":"42ed563af623d57948fd756451470da1c24b3c2a6c53ad66f876cf11f48900c5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-22T13:33:17Z","title_canon_sha256":"213d48ca9e91466d8a64541c1eaf0947af845ff4fd80089e454562f34744145a"},"schema_version":"1.0","source":{"id":"2606.23322","kind":"arxiv","version":1}},"canonical_sha256":"48a53ffb93acb8ea933c6b6dbd9de50071115f777761961f419ffcfccb1cbc03","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48a53ffb93acb8ea933c6b6dbd9de50071115f777761961f419ffcfccb1cbc03","first_computed_at":"2026-06-23T03:14:16.798286Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T03:14:16.798286Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b2ZuMg3cCBiDhr/qTRvhF5J9F1FkFQp6B2pt23QiGLtL3MgzC5cKSKJOat5KyRNJBdfaVLJC/DQl4/1i/n1PBQ==","signature_status":"signed_v1","signed_at":"2026-06-23T03:14:16.798623Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.23322","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e148bd24ae0b9ee9ef197c77e2d963b997cdd6e562c2b13d4410f0429773f068","sha256:2b387719c6cc950def281f8947e230158c5cbbfcbf03ec73311e7daae9b8f52c"],"state_sha256":"0410a8e52be23d642252574fdaab786532fd04c9567e093457850545764aff51"}