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Specifically, what is the largest number of assignments (of Hamming weight $t$) accepted by a $k$-CNF formula that only accepts assignments of weight at least $t$? Among others, we provide the following results:\n  - While an optimal solution is known for $t \\leq n/k$, the problem remains open for $t > n/k$. We formulate a (monotone) version of the problem as an extremal hypergraph problem and show that for $t = n-k$, the problem is exactly the Tur\\'{a}n problem.\n  "},"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":"2412.20493","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2024-12-29T15:15:11Z","cross_cats_sorted":[],"title_canon_sha256":"5e523def5268febdf26f82f7b05509b72d0ff9e57fc862f53cd6dfa56c8936c7","abstract_canon_sha256":"8e32184ed062b2fa9322a11e8eced757056fcc19ebdb6ed401d092bc45111f5c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:55:17.872607Z","signature_b64":"qvqex5O0Kw+8d6NQVqeNsgIp9hXP3bAVHUCsLm+x4PyyjQ0HAGwuy5Z0XKGR+2eRZvmMMjWB9D0LaIKP/iL+Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"356f3168f1482387a8e622f7cead511c7d5e36db9e8266db916f264c4af57b64","last_reissued_at":"2026-07-05T09:55:17.872122Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:55:17.872122Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Extremal Properties of k-CNF: Capturing Threshold Functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Marvin K\\\"unnemann, Mohit Gurumukhani, Ramamohan Paturi","submitted_at":"2024-12-29T15:15:11Z","abstract_excerpt":"We consider a basic question on the expressiveness of $k$-CNF formulas: How well can $k$-CNF formulas capture threshold functions? 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We formulate a (monotone) version of the problem as an extremal hypergraph problem and show that for $t = n-k$, the problem is exactly the Tur\\'{a}n problem.\n  "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.20493","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.20493/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2412.20493","created_at":"2026-07-05T09:55:17.872181+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.20493v1","created_at":"2026-07-05T09:55:17.872181+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.20493","created_at":"2026-07-05T09:55:17.872181+00:00"},{"alias_kind":"pith_short_12","alias_value":"GVXTC2HRJARY","created_at":"2026-07-05T09:55:17.872181+00:00"},{"alias_kind":"pith_short_16","alias_value":"GVXTC2HRJARYPKHG","created_at":"2026-07-05T09:55:17.872181+00:00"},{"alias_kind":"pith_short_8","alias_value":"GVXTC2HR","created_at":"2026-07-05T09:55:17.872181+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/GVXTC2HRJARYPKHGEL345LKRDR","json":"https://pith.science/pith/GVXTC2HRJARYPKHGEL345LKRDR.json","graph_json":"https://pith.science/api/pith-number/GVXTC2HRJARYPKHGEL345LKRDR/graph.json","events_json":"https://pith.science/api/pith-number/GVXTC2HRJARYPKHGEL345LKRDR/events.json","paper":"https://pith.science/paper/GVXTC2HR"},"agent_actions":{"view_html":"https://pith.science/pith/GVXTC2HRJARYPKHGEL345LKRDR","download_json":"https://pith.science/pith/GVXTC2HRJARYPKHGEL345LKRDR.json","view_paper":"https://pith.science/paper/GVXTC2HR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.20493&json=true","fetch_graph":"https://pith.science/api/pith-number/GVXTC2HRJARYPKHGEL345LKRDR/graph.json","fetch_events":"https://pith.science/api/pith-number/GVXTC2HRJARYPKHGEL345LKRDR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GVXTC2HRJARYPKHGEL345LKRDR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GVXTC2HRJARYPKHGEL345LKRDR/action/storage_attestation","attest_author":"https://pith.science/pith/GVXTC2HRJARYPKHGEL345LKRDR/action/author_attestation","sign_citation":"https://pith.science/pith/GVXTC2HRJARYPKHGEL345LKRDR/action/citation_signature","submit_replication":"https://pith.science/pith/GVXTC2HRJARYPKHGEL345LKRDR/action/replication_record"}},"created_at":"2026-07-05T09:55:17.872181+00:00","updated_at":"2026-07-05T09:55:17.872181+00:00"}