{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:P6MP56USYBA475PRZ3J4LTQDT2","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":"19e85aebbbca2d742b65acf4ffd6c6444bba2ab453e95f72d1b937b30adca570","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-01T07:56:42Z","title_canon_sha256":"3edb1ccd181e4854e84b0634f953c9280c436803a1ae8603fbc40b59c1bd76c1"},"schema_version":"1.0","source":{"id":"2608.00505","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.00505","created_at":"2026-08-04T00:37:05Z"},{"alias_kind":"arxiv_version","alias_value":"2608.00505v1","created_at":"2026-08-04T00:37:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.00505","created_at":"2026-08-04T00:37:05Z"},{"alias_kind":"pith_short_12","alias_value":"P6MP56USYBA4","created_at":"2026-08-04T00:37:05Z"},{"alias_kind":"pith_short_16","alias_value":"P6MP56USYBA475PR","created_at":"2026-08-04T00:37:05Z"},{"alias_kind":"pith_short_8","alias_value":"P6MP56US","created_at":"2026-08-04T00:37:05Z"}],"graph_snapshots":[{"event_id":"sha256:722f616b2b74614b5ef47ed72fd341d4ec249ed3ef83438ca685e93323de80ee","target":"graph","created_at":"2026-08-04T00:37:05Z","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/2608.00505/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $V$ be an $n$-dimensional vector space over the finite field $\\mathbb{F}_q$, and ${V\\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\\mathcal{F}\\subseteq {V\\brack k}$ and $\\mathcal{G}\\subseteq {V\\brack \\ell}$ are said to be cross $t$-intersecting if $\\dim(F\\cap G)\\geq t$ for all $F\\in\\mathcal{F}$ and $G\\in \\mathcal{G}$. In this paper, we determine the extremal structures when $|\\mathcal{F}||\\mathcal{G}|$ attains the maximum value under the conditions $\\dim\\left(\\cap_{F\\in \\mathcal{F}}F\\right)<t$ and $\\dim\\left(\\cap_{G\\in \\mathcal{G}}G\\right)<t$.","authors_text":"Benjian Lv, Kaishun Wang, Yu Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-01T07:56:42Z","title":"Extremal cross $t$-intersecting families under $t$-covering number constraints for vector spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.00505","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":"sha256:6b8d619ebbdbc77e7c7fc957e5b505b67d02c94e14d4caac5a3a7bee7556126c","target":"record","created_at":"2026-08-04T00:37:05Z","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":"19e85aebbbca2d742b65acf4ffd6c6444bba2ab453e95f72d1b937b30adca570","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-01T07:56:42Z","title_canon_sha256":"3edb1ccd181e4854e84b0634f953c9280c436803a1ae8603fbc40b59c1bd76c1"},"schema_version":"1.0","source":{"id":"2608.00505","kind":"arxiv","version":1}},"canonical_sha256":"7f98fefa92c041cff5f1ced3c5ce039e9461977196c09c3bcaa0683867f7222a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7f98fefa92c041cff5f1ced3c5ce039e9461977196c09c3bcaa0683867f7222a","first_computed_at":"2026-08-04T00:37:05.191218Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-04T00:37:05.191218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NGNFNv0RoZjD3cQB7a+qa7d4O2b88eAtL8FAF8hvTE4zheAXO0YK7BmzEuCJNnBbAz5Us8B9oqNHuVPpMnNSAg==","signature_status":"signed_v1","signed_at":"2026-08-04T00:37:05.192850Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.00505","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6b8d619ebbdbc77e7c7fc957e5b505b67d02c94e14d4caac5a3a7bee7556126c","sha256:722f616b2b74614b5ef47ed72fd341d4ec249ed3ef83438ca685e93323de80ee"],"state_sha256":"d916eee127474cae5746575ab3f87b25bea4d3016bb58d25e1ad6a8807915191"}