{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:ENYU6CZMV4XK4JFGJJMBBTO3KJ","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":"72109c4ddea2ffb7823cd76d3443c6473d32b49774134285588c91e06ee516f1","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-04-03T17:51:50Z","title_canon_sha256":"72bf6f85c07dbfb92679200557bd5ed46c35d80b65f417c6bb79f8edb416ec1f"},"schema_version":"1.0","source":{"id":"2504.02806","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.02806","created_at":"2026-07-05T11:02:22Z"},{"alias_kind":"arxiv_version","alias_value":"2504.02806v3","created_at":"2026-07-05T11:02:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.02806","created_at":"2026-07-05T11:02:22Z"},{"alias_kind":"pith_short_12","alias_value":"ENYU6CZMV4XK","created_at":"2026-07-05T11:02:22Z"},{"alias_kind":"pith_short_16","alias_value":"ENYU6CZMV4XK4JFG","created_at":"2026-07-05T11:02:22Z"},{"alias_kind":"pith_short_8","alias_value":"ENYU6CZM","created_at":"2026-07-05T11:02:22Z"}],"graph_snapshots":[{"event_id":"sha256:b7cb2ccbe897d597b3c844e0c724b1f96a4cc0923ea6f26cac43858280f06e99","target":"graph","created_at":"2026-07-05T11:02:22Z","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/2504.02806/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Tur\\'{a}n's theorem, if $G$ is $K_{r+1}$-free, then $m \\leq |E(T(n, r))|,$ where $T(n, r)$ denotes the Tur\\'{a}n graph on $n$ vertices with a maximum clique of order $r$. A limitation of this statement is that it does not give an expression in terms of $n$ and $r$. A widely used version of Tur\\'{a}n's theorem states that for an $n$-vertex $K_{r+1}$-free graph, $m \\leq \\left\\lfloor \\frac{n^2(r-1)}{2r} \\right\\rfloor.$ Though this bound is often more convenient, it is not the same as the original statement. In particular, the","authors_text":"Bangalore), L. Sunil Chandran (Indian Institute of Science, Rajat Adak","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-04-03T17:51:50Z","title":"Vertex-Based Localization of Tur\\'{a}n's Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.02806","kind":"arxiv","version":3},"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":"sha256:83a73580f28c5f714370901a6b82852a0b3d62d31e1cc24911733197e849b456","target":"record","created_at":"2026-07-05T11:02:22Z","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":"72109c4ddea2ffb7823cd76d3443c6473d32b49774134285588c91e06ee516f1","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-04-03T17:51:50Z","title_canon_sha256":"72bf6f85c07dbfb92679200557bd5ed46c35d80b65f417c6bb79f8edb416ec1f"},"schema_version":"1.0","source":{"id":"2504.02806","kind":"arxiv","version":3}},"canonical_sha256":"23714f0b2caf2eae24a64a5810cddb524d9a09a10b837e0f054c33b8ed51a413","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"23714f0b2caf2eae24a64a5810cddb524d9a09a10b837e0f054c33b8ed51a413","first_computed_at":"2026-07-05T11:02:22.741046Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:02:22.741046Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BdbxN3V6CpxA8v7JLJUQcUCAkDGfiqNLb54kzBdA+419m58+j4kXxd6DM+9bXguJ8jweR3v71OrQS8bI1XFcBg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:02:22.741560Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.02806","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:83a73580f28c5f714370901a6b82852a0b3d62d31e1cc24911733197e849b456","sha256:b7cb2ccbe897d597b3c844e0c724b1f96a4cc0923ea6f26cac43858280f06e99"],"state_sha256":"f1ec0efdbbdad2dc1918259aea3120dbec24c0fa9df02937690ffa6ef6484450"}