{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:BFJ7X2W3LEMDSRWX2HVXGNICJZ","short_pith_number":"pith:BFJ7X2W3","schema_version":"1.0","canonical_sha256":"0953fbeadb59183946d7d1eb7335024e51d9225451157da36ea3b55a578b5e35","source":{"kind":"arxiv","id":"2606.25931","version":1},"attestation_state":"computed","paper":{"title":"A Simple Counting Argument for Dense Linear Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J\\'ozsef Solymosi, Lior Gishboliner","submitted_at":"2026-06-24T15:07:33Z","abstract_excerpt":"In connection to the Brown-Erd\\H{o}s-S\\'os conjecture, we give a short local averaging proof of a density theorem for linear uniform hypergraphs. Let $r \\ge 3$, $k \\ge 3$, and suppose that $n \\ge (r-2)(k-2)+1$. If $H$ is a linear $r$-uniform hypergraph on $n$ vertices and \\[|E(H)| \\geq \\frac{k-2}{r^2((r-2)(k-2)+1)}n^2 + \\frac{n}{r},\\] then $H$ contains $k$ edges spanning at most $(r-2)k+3$ vertices. In the standard linear-density normalization, this gives the asymptotic density threshold $c \\geq \\frac{r-1}{r} \\cdot \\frac{k-2}{(r-2)(k-2)+1} + o(1)$. In particular, this yields a simple proof of "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.25931","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-24T15:07:33Z","cross_cats_sorted":[],"title_canon_sha256":"17870507339494418c44af118781c163fa07c312589389c34e0d09b690666f68","abstract_canon_sha256":"6ad0b9ab69bf5b343714c4d8c31150b5f47269bf2c557bb2df4d63f96993e063"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-25T01:18:43.490766Z","signature_b64":"rcLKjQwqr42kJpRVrNREZX1aVcu5N8hF3BZKbqFuu+i5Zr0qJY9lc170CFAzXH92OOhca/0/fvLeXEtgOdoXAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0953fbeadb59183946d7d1eb7335024e51d9225451157da36ea3b55a578b5e35","last_reissued_at":"2026-06-25T01:18:43.490424Z","signature_status":"signed_v1","first_computed_at":"2026-06-25T01:18:43.490424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Simple Counting Argument for Dense Linear Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J\\'ozsef Solymosi, Lior Gishboliner","submitted_at":"2026-06-24T15:07:33Z","abstract_excerpt":"In connection to the Brown-Erd\\H{o}s-S\\'os conjecture, we give a short local averaging proof of a density theorem for linear uniform hypergraphs. Let $r \\ge 3$, $k \\ge 3$, and suppose that $n \\ge (r-2)(k-2)+1$. If $H$ is a linear $r$-uniform hypergraph on $n$ vertices and \\[|E(H)| \\geq \\frac{k-2}{r^2((r-2)(k-2)+1)}n^2 + \\frac{n}{r},\\] then $H$ contains $k$ edges spanning at most $(r-2)k+3$ vertices. In the standard linear-density normalization, this gives the asymptotic density threshold $c \\geq \\frac{r-1}{r} \\cdot \\frac{k-2}{(r-2)(k-2)+1} + o(1)$. In particular, this yields a simple proof of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25931","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.25931/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.25931","created_at":"2026-06-25T01:18:43.490486+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.25931v1","created_at":"2026-06-25T01:18:43.490486+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.25931","created_at":"2026-06-25T01:18:43.490486+00:00"},{"alias_kind":"pith_short_12","alias_value":"BFJ7X2W3LEMD","created_at":"2026-06-25T01:18:43.490486+00:00"},{"alias_kind":"pith_short_16","alias_value":"BFJ7X2W3LEMDSRWX","created_at":"2026-06-25T01:18:43.490486+00:00"},{"alias_kind":"pith_short_8","alias_value":"BFJ7X2W3","created_at":"2026-06-25T01:18:43.490486+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ","json":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ.json","graph_json":"https://pith.science/api/pith-number/BFJ7X2W3LEMDSRWX2HVXGNICJZ/graph.json","events_json":"https://pith.science/api/pith-number/BFJ7X2W3LEMDSRWX2HVXGNICJZ/events.json","paper":"https://pith.science/paper/BFJ7X2W3"},"agent_actions":{"view_html":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ","download_json":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ.json","view_paper":"https://pith.science/paper/BFJ7X2W3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.25931&json=true","fetch_graph":"https://pith.science/api/pith-number/BFJ7X2W3LEMDSRWX2HVXGNICJZ/graph.json","fetch_events":"https://pith.science/api/pith-number/BFJ7X2W3LEMDSRWX2HVXGNICJZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/action/storage_attestation","attest_author":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/action/author_attestation","sign_citation":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/action/citation_signature","submit_replication":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/action/replication_record"}},"created_at":"2026-06-25T01:18:43.490486+00:00","updated_at":"2026-06-25T01:18:43.490486+00:00"}