{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:3JYHAI2ZXBX33BWZBKWDUT3FX5","short_pith_number":"pith:3JYHAI2Z","schema_version":"1.0","canonical_sha256":"da70702359b86fbd86d90aac3a4f65bf67164aaa1eda8b5b2d70011e71866a5a","source":{"kind":"arxiv","id":"1409.0081","version":1},"attestation_state":"computed","paper":{"title":"On $k$-Gons and $k$-Holes in Point Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"cs.DM","authors_text":"Birgit Vogtenhuber, Clemens Huemer, Hern\\'an Gonz\\'alez-Aguilar, Jorge Urrutia, Marco A. Heredia, Oswin Aichholzer, Pavel Valtr, Ruy Fabila-Monroy, Thomas Hackl","submitted_at":"2014-08-30T04:21:07Z","abstract_excerpt":"We consider a variation of the classical Erd\\H{o}s-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower bound for the number of $k$-holes, and show that this number is maximized by sets in convex position."},"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":"1409.0081","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-08-30T04:21:07Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"1665516e00ade9e98fc9ec05aa590ad60f883036dca1d5258cb12b831e950a52","abstract_canon_sha256":"f427db3d3e774c702032adcac8d77330af558d4b09b6bf8308928148e2197bfe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:53.422005Z","signature_b64":"hm5OA0ufr5VlEm1YMURjsFxNCEnS9ERSLFOULbQTaCxLC0Eqc4xJzVbadKFw+ErTRkdLDY03WxvtnDOjpy5WBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da70702359b86fbd86d90aac3a4f65bf67164aaa1eda8b5b2d70011e71866a5a","last_reissued_at":"2026-05-18T02:43:53.421641Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:53.421641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On $k$-Gons and $k$-Holes in Point Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"cs.DM","authors_text":"Birgit Vogtenhuber, Clemens Huemer, Hern\\'an Gonz\\'alez-Aguilar, Jorge Urrutia, Marco A. Heredia, Oswin Aichholzer, Pavel Valtr, Ruy Fabila-Monroy, Thomas Hackl","submitted_at":"2014-08-30T04:21:07Z","abstract_excerpt":"We consider a variation of the classical Erd\\H{o}s-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower bound for the number of $k$-holes, and show that this number is maximized by sets in convex position."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0081","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":"1409.0081","created_at":"2026-05-18T02:43:53.421695+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.0081v1","created_at":"2026-05-18T02:43:53.421695+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0081","created_at":"2026-05-18T02:43:53.421695+00:00"},{"alias_kind":"pith_short_12","alias_value":"3JYHAI2ZXBX3","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_16","alias_value":"3JYHAI2ZXBX33BWZ","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_8","alias_value":"3JYHAI2Z","created_at":"2026-05-18T12:28:11.866339+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/3JYHAI2ZXBX33BWZBKWDUT3FX5","json":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5.json","graph_json":"https://pith.science/api/pith-number/3JYHAI2ZXBX33BWZBKWDUT3FX5/graph.json","events_json":"https://pith.science/api/pith-number/3JYHAI2ZXBX33BWZBKWDUT3FX5/events.json","paper":"https://pith.science/paper/3JYHAI2Z"},"agent_actions":{"view_html":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5","download_json":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5.json","view_paper":"https://pith.science/paper/3JYHAI2Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.0081&json=true","fetch_graph":"https://pith.science/api/pith-number/3JYHAI2ZXBX33BWZBKWDUT3FX5/graph.json","fetch_events":"https://pith.science/api/pith-number/3JYHAI2ZXBX33BWZBKWDUT3FX5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/action/storage_attestation","attest_author":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/action/author_attestation","sign_citation":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/action/citation_signature","submit_replication":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/action/replication_record"}},"created_at":"2026-05-18T02:43:53.421695+00:00","updated_at":"2026-05-18T02:43:53.421695+00:00"}