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We prove a labelled first-order limit law, i.e., for every first-order sentence $\\varphi$, the proportion of graphs in $\\mbP_n(l,d)$ that satisfy $\\varphi$ converges as $n \\to \\infty$. By combining this result with a result of Hundack, Pr\\\"omel and Steger \\cite{HPS} we also prove that if $1 \\leq s_1 \\leq ... \\leq s_l$ are integers"},"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":"1204.2454","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-04-11T14:05:04Z","cross_cats_sorted":[],"title_canon_sha256":"4322a9ac9f813a5547d77f2df60ea479622d91a857063328082ad1b071f53417","abstract_canon_sha256":"e0107ceaebd136abe44c079c0b851a7c671f645c542d52cef033afb55f74d2ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:26.119481Z","signature_b64":"Qa4Cl27sy0VzsR7H2ZaTz4X37XnE/4aMSbaCajfJkzTCiGTs26OFbYVVqFmDxRTY+xXJc2D/3y/dvshK8ivaBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8ff925fadc8c23d590cdc3723befdba6ec49257387aeee5c26504a324651ac4","last_reissued_at":"2026-05-18T03:33:26.118587Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:26.118587Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A limit law of almost $l$-partite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Vera Koponen","submitted_at":"2012-04-11T14:05:04Z","abstract_excerpt":"For integers $l \\geq 2$, $d \\geq 1$ we study (undirected) graphs with vertices $1, ..., n$ such that the vertices can be partitioned into $l$ parts such that every vertex has at most $d$ neighbours in its own part. 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By combining this result with a result of Hundack, Pr\\\"omel and Steger \\cite{HPS} we also prove that if $1 \\leq s_1 \\leq ... \\leq s_l$ are integers"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.2454","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":"1204.2454","created_at":"2026-05-18T03:33:26.118741+00:00"},{"alias_kind":"arxiv_version","alias_value":"1204.2454v2","created_at":"2026-05-18T03:33:26.118741+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.2454","created_at":"2026-05-18T03:33:26.118741+00:00"},{"alias_kind":"pith_short_12","alias_value":"VD7ZEX5NZDBD","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"VD7ZEX5NZDBD2WIM","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"VD7ZEX5N","created_at":"2026-05-18T12:27:25.539911+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/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ","json":"https://pith.science/pith/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ.json","graph_json":"https://pith.science/api/pith-number/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ/graph.json","events_json":"https://pith.science/api/pith-number/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ/events.json","paper":"https://pith.science/paper/VD7ZEX5N"},"agent_actions":{"view_html":"https://pith.science/pith/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ","download_json":"https://pith.science/pith/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ.json","view_paper":"https://pith.science/paper/VD7ZEX5N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1204.2454&json=true","fetch_graph":"https://pith.science/api/pith-number/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ/graph.json","fetch_events":"https://pith.science/api/pith-number/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ/action/storage_attestation","attest_author":"https://pith.science/pith/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ/action/author_attestation","sign_citation":"https://pith.science/pith/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ/action/citation_signature","submit_replication":"https://pith.science/pith/VD7ZEX5NZDBD2WIM3Q3SHPX5XJ/action/replication_record"}},"created_at":"2026-05-18T03:33:26.118741+00:00","updated_at":"2026-05-18T03:33:26.118741+00:00"}