{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:EI477JOOTZGFJIOTYMBR33HDL6","short_pith_number":"pith:EI477JOO","schema_version":"1.0","canonical_sha256":"2239ffa5ce9e4c54a1d3c3031dece35f94bb8c533bbacc9b963e1fcd021458fc","source":{"kind":"arxiv","id":"1411.1698","version":2},"attestation_state":"computed","paper":{"title":"On the Max-Cut of Sparse Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"David Gamarnik, Quan Li","submitted_at":"2014-11-06T18:42:41Z","abstract_excerpt":"We consider the problem of estimating the size of a maximum cut (Max-Cut problem) in a random Erd\\H{o}s-R\\'{e}nyi graph on $n$ nodes and $\\lfloor cn \\rfloor$ edges. It is shown in Coppersmith et al. ~\\cite{Coppersmith2004} that the size of the maximum cut in this graph normalized by the number of nodes belongs to the asymptotic region $[c/2+0.37613\\sqrt{c},c/2+0.58870\\sqrt{c}]$ with high probability (w.h.p.) as $n$ increases, for all sufficiently large $c$.\n  In this paper we improve both upper and lower bounds by introducing a novel bounding technique. Specifically, we establish that the size"},"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":"1411.1698","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-11-06T18:42:41Z","cross_cats_sorted":[],"title_canon_sha256":"812799748fd77989954baa5ae23561290e69ddf291777fd23226011a67fd290a","abstract_canon_sha256":"bd7449223549a08491bd25f4283496f5260dd05b5d549dcde98c91fa18ac7bc2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:57.960240Z","signature_b64":"S+zXkSF/P8/l7hwPZ6X7liAZzgYIgZ7DQQcNX5DjfmS677CIOSRJDjGDuyWCxePXR1oxeMrri8hyvZnl9cxiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2239ffa5ce9e4c54a1d3c3031dece35f94bb8c533bbacc9b963e1fcd021458fc","last_reissued_at":"2026-05-18T00:50:57.959856Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:57.959856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Max-Cut of Sparse Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"David Gamarnik, Quan Li","submitted_at":"2014-11-06T18:42:41Z","abstract_excerpt":"We consider the problem of estimating the size of a maximum cut (Max-Cut problem) in a random Erd\\H{o}s-R\\'{e}nyi graph on $n$ nodes and $\\lfloor cn \\rfloor$ edges. It is shown in Coppersmith et al. ~\\cite{Coppersmith2004} that the size of the maximum cut in this graph normalized by the number of nodes belongs to the asymptotic region $[c/2+0.37613\\sqrt{c},c/2+0.58870\\sqrt{c}]$ with high probability (w.h.p.) as $n$ increases, for all sufficiently large $c$.\n  In this paper we improve both upper and lower bounds by introducing a novel bounding technique. Specifically, we establish that the size"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1698","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1411.1698","created_at":"2026-05-18T00:50:57.959913+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.1698v2","created_at":"2026-05-18T00:50:57.959913+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.1698","created_at":"2026-05-18T00:50:57.959913+00:00"},{"alias_kind":"pith_short_12","alias_value":"EI477JOOTZGF","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_16","alias_value":"EI477JOOTZGFJIOT","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_8","alias_value":"EI477JOO","created_at":"2026-05-18T12:28:25.294606+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/EI477JOOTZGFJIOTYMBR33HDL6","json":"https://pith.science/pith/EI477JOOTZGFJIOTYMBR33HDL6.json","graph_json":"https://pith.science/api/pith-number/EI477JOOTZGFJIOTYMBR33HDL6/graph.json","events_json":"https://pith.science/api/pith-number/EI477JOOTZGFJIOTYMBR33HDL6/events.json","paper":"https://pith.science/paper/EI477JOO"},"agent_actions":{"view_html":"https://pith.science/pith/EI477JOOTZGFJIOTYMBR33HDL6","download_json":"https://pith.science/pith/EI477JOOTZGFJIOTYMBR33HDL6.json","view_paper":"https://pith.science/paper/EI477JOO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.1698&json=true","fetch_graph":"https://pith.science/api/pith-number/EI477JOOTZGFJIOTYMBR33HDL6/graph.json","fetch_events":"https://pith.science/api/pith-number/EI477JOOTZGFJIOTYMBR33HDL6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EI477JOOTZGFJIOTYMBR33HDL6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EI477JOOTZGFJIOTYMBR33HDL6/action/storage_attestation","attest_author":"https://pith.science/pith/EI477JOOTZGFJIOTYMBR33HDL6/action/author_attestation","sign_citation":"https://pith.science/pith/EI477JOOTZGFJIOTYMBR33HDL6/action/citation_signature","submit_replication":"https://pith.science/pith/EI477JOOTZGFJIOTYMBR33HDL6/action/replication_record"}},"created_at":"2026-05-18T00:50:57.959913+00:00","updated_at":"2026-05-18T00:50:57.959913+00:00"}