{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:BFJ7X2W3LEMDSRWX2HVXGNICJZ","short_pith_number":"pith:BFJ7X2W3","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"},"canonical_sha256":"0953fbeadb59183946d7d1eb7335024e51d9225451157da36ea3b55a578b5e35","source":{"kind":"arxiv","id":"2606.25931","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.25931","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"arxiv_version","alias_value":"2606.25931v1","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.25931","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"pith_short_12","alias_value":"BFJ7X2W3LEMD","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"pith_short_16","alias_value":"BFJ7X2W3LEMDSRWX","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"pith_short_8","alias_value":"BFJ7X2W3","created_at":"2026-06-25T01:18:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:BFJ7X2W3LEMDSRWX2HVXGNICJZ","target":"record","payload":{"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"},"canonical_sha256":"0953fbeadb59183946d7d1eb7335024e51d9225451157da36ea3b55a578b5e35","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"},"source_kind":"arxiv","source_id":"2606.25931","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-25T01:18:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fOUnkRs4uSNq0AnSbGJmXpKfmsQdBvn1z5jyKlHBOa1xXiACvfU3M+Q5OQ7EoURTO0j863+qLItIfdw7EfTJBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T09:41:50.180905Z"},"content_sha256":"3c1d2cf973ae33cac3569d34aa424ea8a2d78792ce62fdafd235b90ace3939aa","schema_version":"1.0","event_id":"sha256:3c1d2cf973ae33cac3569d34aa424ea8a2d78792ce62fdafd235b90ace3939aa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:BFJ7X2W3LEMDSRWX2HVXGNICJZ","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-25T01:18:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YiMVIxCiSt4DtxXC4j8yet4sQCE6vjsxScaXvnoMurZlYphU2N0PeWKOLZpavMKiv03YzRe1FCz6ks0c8t3/CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T09:41:50.181327Z"},"content_sha256":"32248d5b06904ca3ca9d22bdcd658c502a64017cb312522acaa012c9bef93432","schema_version":"1.0","event_id":"sha256:32248d5b06904ca3ca9d22bdcd658c502a64017cb312522acaa012c9bef93432"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/bundle.json","state_url":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-06T09:41:50Z","links":{"resolver":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ","bundle":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/bundle.json","state":"https://pith.science/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BFJ7X2W3LEMDSRWX2HVXGNICJZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:BFJ7X2W3LEMDSRWX2HVXGNICJZ","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":"6ad0b9ab69bf5b343714c4d8c31150b5f47269bf2c557bb2df4d63f96993e063","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-24T15:07:33Z","title_canon_sha256":"17870507339494418c44af118781c163fa07c312589389c34e0d09b690666f68"},"schema_version":"1.0","source":{"id":"2606.25931","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.25931","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"arxiv_version","alias_value":"2606.25931v1","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.25931","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"pith_short_12","alias_value":"BFJ7X2W3LEMD","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"pith_short_16","alias_value":"BFJ7X2W3LEMDSRWX","created_at":"2026-06-25T01:18:43Z"},{"alias_kind":"pith_short_8","alias_value":"BFJ7X2W3","created_at":"2026-06-25T01:18:43Z"}],"graph_snapshots":[{"event_id":"sha256:32248d5b06904ca3ca9d22bdcd658c502a64017cb312522acaa012c9bef93432","target":"graph","created_at":"2026-06-25T01:18:43Z","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/2606.25931/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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 ","authors_text":"J\\'ozsef Solymosi, Lior Gishboliner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-24T15:07:33Z","title":"A Simple Counting Argument for Dense Linear Hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25931","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":"sha256:3c1d2cf973ae33cac3569d34aa424ea8a2d78792ce62fdafd235b90ace3939aa","target":"record","created_at":"2026-06-25T01:18:43Z","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":"6ad0b9ab69bf5b343714c4d8c31150b5f47269bf2c557bb2df4d63f96993e063","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-24T15:07:33Z","title_canon_sha256":"17870507339494418c44af118781c163fa07c312589389c34e0d09b690666f68"},"schema_version":"1.0","source":{"id":"2606.25931","kind":"arxiv","version":1}},"canonical_sha256":"0953fbeadb59183946d7d1eb7335024e51d9225451157da36ea3b55a578b5e35","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0953fbeadb59183946d7d1eb7335024e51d9225451157da36ea3b55a578b5e35","first_computed_at":"2026-06-25T01:18:43.490424Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-25T01:18:43.490424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rcLKjQwqr42kJpRVrNREZX1aVcu5N8hF3BZKbqFuu+i5Zr0qJY9lc170CFAzXH92OOhca/0/fvLeXEtgOdoXAQ==","signature_status":"signed_v1","signed_at":"2026-06-25T01:18:43.490766Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.25931","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c1d2cf973ae33cac3569d34aa424ea8a2d78792ce62fdafd235b90ace3939aa","sha256:32248d5b06904ca3ca9d22bdcd658c502a64017cb312522acaa012c9bef93432"],"state_sha256":"d06faeec3c42e43050c0374cb6596b635e3409c5cbb69a6ddca5b77dcd505f95"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"48P0wned0SbABCdZc+2hwo1+GvLdWBNQc1btJ80IDe+hEi7HtJa2EZ2OOKVfPrfeCF07uVejEO/Tu+2JXoLaBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T09:41:50.183622Z","bundle_sha256":"f4516a1678b897efea110bb1c39a8c68dde659ee08fed29ede68f195db26c152"}}