{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:GX4J77RO7TSPGXATS4E2QGP5M6","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":"f205941622bcda8a4d983dfaee892a1b9c42942cff039c2eb517aaa2d135c65f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-12T15:29:35Z","title_canon_sha256":"f608e85997ed56837487d841d1361960f7ade1811ff7a199feec2f2a87689d4f"},"schema_version":"1.0","source":{"id":"1405.2805","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.2805","created_at":"2026-05-18T02:28:20Z"},{"alias_kind":"arxiv_version","alias_value":"1405.2805v2","created_at":"2026-05-18T02:28:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.2805","created_at":"2026-05-18T02:28:20Z"},{"alias_kind":"pith_short_12","alias_value":"GX4J77RO7TSP","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GX4J77RO7TSPGXAT","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GX4J77RO","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:b7475d383078dd29785232068c9cf988eabf3cd341dca8e3a96a8c6036ebe405","target":"graph","created_at":"2026-05-18T02:28:20Z","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":"Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \\le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \\subseteq S_p$, are said to be $r$-cross-intersecting if no matter how we select $x \\in A$ and $y \\in B$, there are at least $r$ distinct indices $i$ such that $x_i = y_i$. We determine the maximum value of $|A|\\cdot|B|$ over all pairs of $r$- cross-intersecting families and characterize the extremal pairs for $r \\ge 1$, provided that $\\min p_i >r+1$. The case $\\min ","authors_text":"G\\'abor Tardos, J\\'anos Pach","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-12T15:29:35Z","title":"Cross-intersecting families of vectors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2805","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":"sha256:7f34a4e1871ac259dbb53b68bc3bb55966c40261cdbc44ca060b11d3be4f1017","target":"record","created_at":"2026-05-18T02:28:20Z","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":"f205941622bcda8a4d983dfaee892a1b9c42942cff039c2eb517aaa2d135c65f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-12T15:29:35Z","title_canon_sha256":"f608e85997ed56837487d841d1361960f7ade1811ff7a199feec2f2a87689d4f"},"schema_version":"1.0","source":{"id":"1405.2805","kind":"arxiv","version":2}},"canonical_sha256":"35f89ffe2efce4f35c139709a819fd67a972dbb9969eb4d15d1bc164dd7e6fb7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"35f89ffe2efce4f35c139709a819fd67a972dbb9969eb4d15d1bc164dd7e6fb7","first_computed_at":"2026-05-18T02:28:20.831302Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:20.831302Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TW0UN+XTAhx3hMihXVBNRjUEM1TMvm1M0jojNQupmeUQ3p5DKfn7IalFY91Z1kV6ZawCUJIzRYa4I+r3gupgAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:20.831844Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.2805","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7f34a4e1871ac259dbb53b68bc3bb55966c40261cdbc44ca060b11d3be4f1017","sha256:b7475d383078dd29785232068c9cf988eabf3cd341dca8e3a96a8c6036ebe405"],"state_sha256":"cb95883164114a2b5c150c71ebd5a56108bfc0ad6aace9c3e3e103fc6193e8d8"}