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If $p\\binom{n}{2}$ is a non-negative integer, define \\[ f(n,p) = \\min\\{f(G) : \\vert V(G)\\vert = n, \\ \\vert E(G)\\vert = p\\binom{n}{2} \\}.\\] Erd\\H{o}s, \\L uczak and Spencer proved that for $n \\geq 2$, \\[ (2n)^{\\frac{1}{2}} - 2 \\leq f(n, {\\frac{1}{2}}) \\leq 4n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}}.\\] In this paper, we prove the following "},"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":"1505.03072","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-12T15:58:41Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"948a5d4fc25202f42769a0a8cd960ac38a1f0f22248c32c71ecab163e0a100ec","abstract_canon_sha256":"e49afb80fd1d369aac90b348f95809de1a014c8289d6af4868bc76ce6b271a7a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:42.387436Z","signature_b64":"V3N/5917sc5c4jbmzWTKt7hxoK1s1hPvrUnC6fCi86q+SUy6aON3YUxhdP9Cp5w7WMOpZLkVkdcx6QFjVwhjBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50c2d823ff1ae9927293c2f6fb8b08222c779a0086c98edf840199432912c4ab","last_reissued_at":"2026-05-18T01:01:42.386876Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:42.386876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Full subgraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jacques Verstra\\\"ete, Klas Markstr\\\"om, Victor Falgas-Ravry","submitted_at":"2015-05-12T15:58:41Z","abstract_excerpt":"Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erd\\H{o}s, \\L uczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a largest full subgraph of $G$. If $p\\binom{n}{2}$ is a non-negative integer, define \\[ f(n,p) = \\min\\{f(G) : \\vert V(G)\\vert = n, \\ \\vert E(G)\\vert = p\\binom{n}{2} \\}.\\] Erd\\H{o}s, \\L uczak and Spencer proved that for $n \\geq 2$, \\[ (2n)^{\\frac{1}{2}} - 2 \\leq f(n, {\\frac{1}{2}}) \\leq 4n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}}.\\] In this paper, we prove the following "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03072","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":"1505.03072","created_at":"2026-05-18T01:01:42.386967+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.03072v2","created_at":"2026-05-18T01:01:42.386967+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03072","created_at":"2026-05-18T01:01:42.386967+00:00"},{"alias_kind":"pith_short_12","alias_value":"KDBNQI77DLUZ","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"KDBNQI77DLUZE4UT","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"KDBNQI77","created_at":"2026-05-18T12:29:27.538025+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/KDBNQI77DLUZE4UTYL3PXCYIEI","json":"https://pith.science/pith/KDBNQI77DLUZE4UTYL3PXCYIEI.json","graph_json":"https://pith.science/api/pith-number/KDBNQI77DLUZE4UTYL3PXCYIEI/graph.json","events_json":"https://pith.science/api/pith-number/KDBNQI77DLUZE4UTYL3PXCYIEI/events.json","paper":"https://pith.science/paper/KDBNQI77"},"agent_actions":{"view_html":"https://pith.science/pith/KDBNQI77DLUZE4UTYL3PXCYIEI","download_json":"https://pith.science/pith/KDBNQI77DLUZE4UTYL3PXCYIEI.json","view_paper":"https://pith.science/paper/KDBNQI77","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.03072&json=true","fetch_graph":"https://pith.science/api/pith-number/KDBNQI77DLUZE4UTYL3PXCYIEI/graph.json","fetch_events":"https://pith.science/api/pith-number/KDBNQI77DLUZE4UTYL3PXCYIEI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KDBNQI77DLUZE4UTYL3PXCYIEI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KDBNQI77DLUZE4UTYL3PXCYIEI/action/storage_attestation","attest_author":"https://pith.science/pith/KDBNQI77DLUZE4UTYL3PXCYIEI/action/author_attestation","sign_citation":"https://pith.science/pith/KDBNQI77DLUZE4UTYL3PXCYIEI/action/citation_signature","submit_replication":"https://pith.science/pith/KDBNQI77DLUZE4UTYL3PXCYIEI/action/replication_record"}},"created_at":"2026-05-18T01:01:42.386967+00:00","updated_at":"2026-05-18T01:01:42.386967+00:00"}