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We show that if $\\mathcal{F}$ is a family of compact, convex sets that does not contain a $C(k)$, then there are $k-2$ lines that pierce $\\mathcal{F}$. Additionally, we give an example of a family of compact, convex sets that contains no $C(k)$ and cannot be pierced by $\\left\\lceil \\frac{k}{2} \\right\\rceil -1$ lines."},"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":"2204.10490","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2022-04-22T04:12:27Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"e5921f5ead6ba4b7690383015ad235ef9ef2a0e5b9521587e1cfb2fc04d74ae9","abstract_canon_sha256":"65c07db7fcd5a9fa4e8ba0cc9e51d8a3f4fd87fe6223331808d1208f4cf809ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:16:57.607500Z","signature_b64":"Uax76U5Y9SaiT2YInapTjxMiG3ybt2VcTDbRN4L47RVkl+87/0o4qMCBWOijrGOxAm+SI1e0FWzsam4aoTnIAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3fe1c268d03fab778d0c223ec9eff5e5690876d1e2c5b036ec74a2038e3f1aac","last_reissued_at":"2026-07-05T04:16:57.607012Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:16:57.607012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Piercing families of convex sets in the plane that avoid a certain subfamily with lines","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Daniel McGinnis","submitted_at":"2022-04-22T04:12:27Z","abstract_excerpt":"We define a $C(k)$ to be a family of $k$ sets $F_1,\\dots,F_k$ such that $\\textrm{conv}(F_i\\cup F_{i+1})\\cap \\textrm{conv}(F_j\\cup F_{j+1})=\\emptyset$ when $\\{i,i+1\\}\\cap \\{j,j+1\\}=\\emptyset$ (indices are taken modulo $k$). We show that if $\\mathcal{F}$ is a family of compact, convex sets that does not contain a $C(k)$, then there are $k-2$ lines that pierce $\\mathcal{F}$. Additionally, we give an example of a family of compact, convex sets that contains no $C(k)$ and cannot be pierced by $\\left\\lceil \\frac{k}{2} \\right\\rceil -1$ lines."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.10490","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/2204.10490/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":"2204.10490","created_at":"2026-07-05T04:16:57.607086+00:00"},{"alias_kind":"arxiv_version","alias_value":"2204.10490v1","created_at":"2026-07-05T04:16:57.607086+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.10490","created_at":"2026-07-05T04:16:57.607086+00:00"},{"alias_kind":"pith_short_12","alias_value":"H7Q4E2GQH6VX","created_at":"2026-07-05T04:16:57.607086+00:00"},{"alias_kind":"pith_short_16","alias_value":"H7Q4E2GQH6VXPDIM","created_at":"2026-07-05T04:16:57.607086+00:00"},{"alias_kind":"pith_short_8","alias_value":"H7Q4E2GQ","created_at":"2026-07-05T04:16:57.607086+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/H7Q4E2GQH6VXPDIMEI7MT37V4V","json":"https://pith.science/pith/H7Q4E2GQH6VXPDIMEI7MT37V4V.json","graph_json":"https://pith.science/api/pith-number/H7Q4E2GQH6VXPDIMEI7MT37V4V/graph.json","events_json":"https://pith.science/api/pith-number/H7Q4E2GQH6VXPDIMEI7MT37V4V/events.json","paper":"https://pith.science/paper/H7Q4E2GQ"},"agent_actions":{"view_html":"https://pith.science/pith/H7Q4E2GQH6VXPDIMEI7MT37V4V","download_json":"https://pith.science/pith/H7Q4E2GQH6VXPDIMEI7MT37V4V.json","view_paper":"https://pith.science/paper/H7Q4E2GQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2204.10490&json=true","fetch_graph":"https://pith.science/api/pith-number/H7Q4E2GQH6VXPDIMEI7MT37V4V/graph.json","fetch_events":"https://pith.science/api/pith-number/H7Q4E2GQH6VXPDIMEI7MT37V4V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H7Q4E2GQH6VXPDIMEI7MT37V4V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H7Q4E2GQH6VXPDIMEI7MT37V4V/action/storage_attestation","attest_author":"https://pith.science/pith/H7Q4E2GQH6VXPDIMEI7MT37V4V/action/author_attestation","sign_citation":"https://pith.science/pith/H7Q4E2GQH6VXPDIMEI7MT37V4V/action/citation_signature","submit_replication":"https://pith.science/pith/H7Q4E2GQH6VXPDIMEI7MT37V4V/action/replication_record"}},"created_at":"2026-07-05T04:16:57.607086+00:00","updated_at":"2026-07-05T04:16:57.607086+00:00"}