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We show that each $m$-thin subset of a group $G$ of cardinality $\\aleph_n$, $n= 0,1,...$ can be partitioned into $\\leqslant m^{n+1}$ 1-thin subsets. On the other side, we construct a group $G$ of cardinality $\\aleph_\\omega$ and point out a 2-thin subset of $G$ which cannot be finitely partitioned into 1-thin subsets."},"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":"1308.1497","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-07T08:20:18Z","cross_cats_sorted":[],"title_canon_sha256":"364d18ea5eb783914f7cd2d2b14a0d756e448238a4b68a9e80426017aa0dc354","abstract_canon_sha256":"40000ec9f155293b08ae80683f402405187b6b9fc7c0a85c31fd82064527bfd6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:37.905181Z","signature_b64":"cefh6V8t6O8DRLGkMDs1GH8wCqREYeJwWdSMuoqoo2m9KQAYfPebZnpPqWd01AuRvH0pOjDAfRPe4yotzlAPCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d70e7aa294629304811f89488fe41b550edebbd276aaba79bcdb564ab1b5cc4","last_reissued_at":"2026-05-18T03:16:37.904627Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:37.904627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Thin subsets of groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"I.V. 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On the other side, we construct a group $G$ of cardinality $\\aleph_\\omega$ and point out a 2-thin subset of $G$ which cannot be finitely partitioned into 1-thin subsets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.1497","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":""},"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":"1308.1497","created_at":"2026-05-18T03:16:37.904709+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.1497v1","created_at":"2026-05-18T03:16:37.904709+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.1497","created_at":"2026-05-18T03:16:37.904709+00:00"},{"alias_kind":"pith_short_12","alias_value":"FVYOPKRJIYUT","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_16","alias_value":"FVYOPKRJIYUTASAR","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_8","alias_value":"FVYOPKRJ","created_at":"2026-05-18T12:27:45.050594+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/FVYOPKRJIYUTASAR7CKIR7SBWV","json":"https://pith.science/pith/FVYOPKRJIYUTASAR7CKIR7SBWV.json","graph_json":"https://pith.science/api/pith-number/FVYOPKRJIYUTASAR7CKIR7SBWV/graph.json","events_json":"https://pith.science/api/pith-number/FVYOPKRJIYUTASAR7CKIR7SBWV/events.json","paper":"https://pith.science/paper/FVYOPKRJ"},"agent_actions":{"view_html":"https://pith.science/pith/FVYOPKRJIYUTASAR7CKIR7SBWV","download_json":"https://pith.science/pith/FVYOPKRJIYUTASAR7CKIR7SBWV.json","view_paper":"https://pith.science/paper/FVYOPKRJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.1497&json=true","fetch_graph":"https://pith.science/api/pith-number/FVYOPKRJIYUTASAR7CKIR7SBWV/graph.json","fetch_events":"https://pith.science/api/pith-number/FVYOPKRJIYUTASAR7CKIR7SBWV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FVYOPKRJIYUTASAR7CKIR7SBWV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FVYOPKRJIYUTASAR7CKIR7SBWV/action/storage_attestation","attest_author":"https://pith.science/pith/FVYOPKRJIYUTASAR7CKIR7SBWV/action/author_attestation","sign_citation":"https://pith.science/pith/FVYOPKRJIYUTASAR7CKIR7SBWV/action/citation_signature","submit_replication":"https://pith.science/pith/FVYOPKRJIYUTASAR7CKIR7SBWV/action/replication_record"}},"created_at":"2026-05-18T03:16:37.904709+00:00","updated_at":"2026-05-18T03:16:37.904709+00:00"}