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Also, we discuss the associated variants \\begin{align*} \\mathrm{non}^*(\\mathcal{I},+)=&\\min\\big\\{|\\mathcal{Y}|:\\mathcal{Y}\\subseteq\\mathcal{I}^+,\\; \\forall\\;A\\in\\mathcal{I}\\;\\exists\\;Y\\in\\mathcal{Y}\\;|A\\cap Y|<\\omega\\big\\}\\\\ \\mathrm{cov}^*(\\mathcal{I},+)=&\\min\\big\\{|\\mathcal{C}|:\\mathcal{C}\\subseteq\\mathcal{I},\\; \\forall\\;Y\\in\\mathcal{I}^+\\;\\exists\\;C\\in\\m"},"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":"1812.01480","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-12-04T15:21:36Z","cross_cats_sorted":[],"title_canon_sha256":"a1a651279120b0f99c8e80a958a1393ac71a2ebe12d352f9afb96555df67ac7c","abstract_canon_sha256":"b1cd7e9453ef30c38fb4c9d01916e8281f9132f699695fb6ff3141891a0af08b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:12.349052Z","signature_b64":"jKt16OeGBaUpalPyfE5QapJpRKqJ4kr6Mgb/Es57LroGVhxX/C5XYIKgLKVcbghe/QjuaRuYeX8y21j5hzlgCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ef48a66eab9e27a2081eec12d9ce51eeb8b02b389ba622b7c71937e6adc0a79","last_reissued_at":"2026-05-17T23:59:12.348616Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:12.348616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ways of Destruction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Barnabas Farkas, Lyubomyr Zdomskyy","submitted_at":"2018-12-04T15:21:36Z","abstract_excerpt":"We study the following natural strong variant of destroying Borel ideals: $\\mathbb{P}$ $\\textit{$+$-destroys}$ $\\mathcal{I}$ if $\\mathbb{P}$ adds an $\\mathcal{I}$-positive set which has finite intersection with every $A\\in\\mathcal{I}\\cap V$. Also, we discuss the associated variants \\begin{align*} \\mathrm{non}^*(\\mathcal{I},+)=&\\min\\big\\{|\\mathcal{Y}|:\\mathcal{Y}\\subseteq\\mathcal{I}^+,\\; \\forall\\;A\\in\\mathcal{I}\\;\\exists\\;Y\\in\\mathcal{Y}\\;|A\\cap Y|<\\omega\\big\\}\\\\ \\mathrm{cov}^*(\\mathcal{I},+)=&\\min\\big\\{|\\mathcal{C}|:\\mathcal{C}\\subseteq\\mathcal{I},\\; \\forall\\;Y\\in\\mathcal{I}^+\\;\\exists\\;C\\in\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.01480","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":"1812.01480","created_at":"2026-05-17T23:59:12.348678+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.01480v1","created_at":"2026-05-17T23:59:12.348678+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.01480","created_at":"2026-05-17T23:59:12.348678+00:00"},{"alias_kind":"pith_short_12","alias_value":"T32IUZXKXHRH","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"T32IUZXKXHRHUIEB","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"T32IUZXK","created_at":"2026-05-18T12:32:53.628368+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/T32IUZXKXHRHUIEB53AS3HHFD3","json":"https://pith.science/pith/T32IUZXKXHRHUIEB53AS3HHFD3.json","graph_json":"https://pith.science/api/pith-number/T32IUZXKXHRHUIEB53AS3HHFD3/graph.json","events_json":"https://pith.science/api/pith-number/T32IUZXKXHRHUIEB53AS3HHFD3/events.json","paper":"https://pith.science/paper/T32IUZXK"},"agent_actions":{"view_html":"https://pith.science/pith/T32IUZXKXHRHUIEB53AS3HHFD3","download_json":"https://pith.science/pith/T32IUZXKXHRHUIEB53AS3HHFD3.json","view_paper":"https://pith.science/paper/T32IUZXK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.01480&json=true","fetch_graph":"https://pith.science/api/pith-number/T32IUZXKXHRHUIEB53AS3HHFD3/graph.json","fetch_events":"https://pith.science/api/pith-number/T32IUZXKXHRHUIEB53AS3HHFD3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T32IUZXKXHRHUIEB53AS3HHFD3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T32IUZXKXHRHUIEB53AS3HHFD3/action/storage_attestation","attest_author":"https://pith.science/pith/T32IUZXKXHRHUIEB53AS3HHFD3/action/author_attestation","sign_citation":"https://pith.science/pith/T32IUZXKXHRHUIEB53AS3HHFD3/action/citation_signature","submit_replication":"https://pith.science/pith/T32IUZXKXHRHUIEB53AS3HHFD3/action/replication_record"}},"created_at":"2026-05-17T23:59:12.348678+00:00","updated_at":"2026-05-17T23:59:12.348678+00:00"}