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The cardinal $\\mathfrak q_0$ is defined as the smallest cardinality $|A|$ of a subset $A\\subset \\mathbb R$ which is not a $Q$-set (a subspace $A\\subset\\mathbb R$ is called a $Q$-set if each subset $B\\subset A$ is of type $F_\\sigma$ in $A$). We present a simple proof of a folklore fact that $\\mathfrak p\\le\\mathfrak q_0\\le\\min\\{\\mathfrak b,\\mathrm{non}(\\mathcal N),\\log(\\mathfrak c^+)\\}$, and also establish the consistency of a number of strict inequalities between the cardin"},"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":"1306.0204","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2013-06-02T12:41:52Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"30f48f80c396cd0a1fcbde28c3c6d6e000d47320794826fb2e877cade05355d8","abstract_canon_sha256":"01da2337d3c1fbddd8acc0da48c133b3e52c97f9d79703b27feaa36a43ec3e73"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:18.030085Z","signature_b64":"uUbmyruq4uR7Z15uZXb644wn7jFxauJT8iMSwHoHUjakKiJLcpjAmQYZm4J9ruFSB/vRxiXd49YrpNiyXAaoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac68a038c92b1c7006bad2aab7ae817d7f6037e8047b9ee5fd1b75e91da25db7","last_reissued_at":"2026-05-18T01:20:18.029631Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:18.029631Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On critical cardinalities related to $Q$-sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.LO","authors_text":"Lubomyr Zdomskyy, Michal Machura, Taras Banakh","submitted_at":"2013-06-02T12:41:52Z","abstract_excerpt":"In this note we collect some known information and prove new results about the small uncountable cardinal $\\mathfrak q_0$. 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