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We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\\aleph_1=a<a^+(\\aleph_1)=c$ and $a=a^+(\\aleph_1)<c$ are both consistent.\n  We also give a several constructions of mad families with some a"},"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":"math/0611744","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2006-11-24T07:50:08Z","cross_cats_sorted":[],"title_canon_sha256":"ca92017309de6db9f38ab0f4873d306aff68b5210f5a1e7f3c4ae8d4e5b09794","abstract_canon_sha256":"4fce5f41ae91be97169740dcef6f827b5c83b18443347fb751e7b91854ddcc07"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:19.718588Z","signature_b64":"4IHE35j/OOC943cDzR7fNS6s3OMjtgZPKnv+QHyoGLm1nucdKqBdUQmQKK5havEz1k/6mabHh4R3puFNdTCfAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d8db80899ddc9aef340627977fe607b82dddff8a2f32b18f918e317e8e73684","last_reissued_at":"2026-05-18T00:49:19.718087Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:19.718087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"How to drive our families mad","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Lajos Soukup, Osvaldo Guzman, Saka\\'e Fuchino, Stefan Geschke","submitted_at":"2006-11-24T07:50:08Z","abstract_excerpt":"Given a family $F$ of pairwise almost disjoint sets on a countable set $S$, we study maximal almost disjoint (mad) families $F^+$ extending $F$.\n  We define $a^+(F)$ to be the minimal possible cardinality of $F^+\\setminus F$ for such $F^+$, and $a^+(\\kappa)=\\sup\\{a^+(F): |F| \\leq \\kappa \\}$. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\\aleph_1=a<a^+(\\aleph_1)=c$ and $a=a^+(\\aleph_1)<c$ are both consistent.\n  We also give a several constructions of mad families with some a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611744","kind":"arxiv","version":3},"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":"math/0611744","created_at":"2026-05-18T00:49:19.718166+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0611744v3","created_at":"2026-05-18T00:49:19.718166+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611744","created_at":"2026-05-18T00:49:19.718166+00:00"},{"alias_kind":"pith_short_12","alias_value":"LWG3QCEZ3XE2","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"LWG3QCEZ3XE2542A","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"LWG3QCEZ","created_at":"2026-05-18T12:25:54.717736+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/LWG3QCEZ3XE2542AMJ4XP7TAPO","json":"https://pith.science/pith/LWG3QCEZ3XE2542AMJ4XP7TAPO.json","graph_json":"https://pith.science/api/pith-number/LWG3QCEZ3XE2542AMJ4XP7TAPO/graph.json","events_json":"https://pith.science/api/pith-number/LWG3QCEZ3XE2542AMJ4XP7TAPO/events.json","paper":"https://pith.science/paper/LWG3QCEZ"},"agent_actions":{"view_html":"https://pith.science/pith/LWG3QCEZ3XE2542AMJ4XP7TAPO","download_json":"https://pith.science/pith/LWG3QCEZ3XE2542AMJ4XP7TAPO.json","view_paper":"https://pith.science/paper/LWG3QCEZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0611744&json=true","fetch_graph":"https://pith.science/api/pith-number/LWG3QCEZ3XE2542AMJ4XP7TAPO/graph.json","fetch_events":"https://pith.science/api/pith-number/LWG3QCEZ3XE2542AMJ4XP7TAPO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LWG3QCEZ3XE2542AMJ4XP7TAPO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LWG3QCEZ3XE2542AMJ4XP7TAPO/action/storage_attestation","attest_author":"https://pith.science/pith/LWG3QCEZ3XE2542AMJ4XP7TAPO/action/author_attestation","sign_citation":"https://pith.science/pith/LWG3QCEZ3XE2542AMJ4XP7TAPO/action/citation_signature","submit_replication":"https://pith.science/pith/LWG3QCEZ3XE2542AMJ4XP7TAPO/action/replication_record"}},"created_at":"2026-05-18T00:49:19.718166+00:00","updated_at":"2026-05-18T00:49:19.718166+00:00"}