{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:3JYHAI2ZXBX33BWZBKWDUT3FX5","short_pith_number":"pith:3JYHAI2Z","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"},"canonical_sha256":"da70702359b86fbd86d90aac3a4f65bf67164aaa1eda8b5b2d70011e71866a5a","source":{"kind":"arxiv","id":"1409.0081","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0081","created_at":"2026-05-18T02:43:53Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0081v1","created_at":"2026-05-18T02:43:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0081","created_at":"2026-05-18T02:43:53Z"},{"alias_kind":"pith_short_12","alias_value":"3JYHAI2ZXBX3","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3JYHAI2ZXBX33BWZ","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3JYHAI2Z","created_at":"2026-05-18T12:28:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:3JYHAI2ZXBX33BWZBKWDUT3FX5","target":"record","payload":{"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"},"canonical_sha256":"da70702359b86fbd86d90aac3a4f65bf67164aaa1eda8b5b2d70011e71866a5a","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"},"source_kind":"arxiv","source_id":"1409.0081","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:43:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lnYN4ALykTj6qF7o70TIZ/+XmlB+PG6/AIS2D4T1p6mJh4jQ2WJx4nsJRCcC5CmhSdtoFlewc3C6A0/xQ83SDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:20:30.182793Z"},"content_sha256":"617778144ee5f8be73e7a9984228257e7bdd4f2f48cfe25f2b32270535d1010d","schema_version":"1.0","event_id":"sha256:617778144ee5f8be73e7a9984228257e7bdd4f2f48cfe25f2b32270535d1010d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:3JYHAI2ZXBX33BWZBKWDUT3FX5","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:43:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fUqM54QRXezzlwuocNdrJHLq6wbO1ss/RAz/lzmPTmW+MRFOtXppcEHm/b/D+Se5tsrq853CJ6Ros/RWcQR/DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:20:30.183463Z"},"content_sha256":"8eb7158a96a1e69c55865f586e0c8ac180c433b52201691644c885e704711658","schema_version":"1.0","event_id":"sha256:8eb7158a96a1e69c55865f586e0c8ac180c433b52201691644c885e704711658"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/bundle.json","state_url":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T17:20:30Z","links":{"resolver":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5","bundle":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/bundle.json","state":"https://pith.science/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3JYHAI2ZXBX33BWZBKWDUT3FX5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:3JYHAI2ZXBX33BWZBKWDUT3FX5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f427db3d3e774c702032adcac8d77330af558d4b09b6bf8308928148e2197bfe","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-08-30T04:21:07Z","title_canon_sha256":"1665516e00ade9e98fc9ec05aa590ad60f883036dca1d5258cb12b831e950a52"},"schema_version":"1.0","source":{"id":"1409.0081","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0081","created_at":"2026-05-18T02:43:53Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0081v1","created_at":"2026-05-18T02:43:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0081","created_at":"2026-05-18T02:43:53Z"},{"alias_kind":"pith_short_12","alias_value":"3JYHAI2ZXBX3","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3JYHAI2ZXBX33BWZ","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3JYHAI2Z","created_at":"2026-05-18T12:28:11Z"}],"graph_snapshots":[{"event_id":"sha256:8eb7158a96a1e69c55865f586e0c8ac180c433b52201691644c885e704711658","target":"graph","created_at":"2026-05-18T02:43:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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.","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","cross_cats":["cs.CG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-08-30T04:21:07Z","title":"On $k$-Gons and $k$-Holes in Point Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0081","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:617778144ee5f8be73e7a9984228257e7bdd4f2f48cfe25f2b32270535d1010d","target":"record","created_at":"2026-05-18T02:43:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f427db3d3e774c702032adcac8d77330af558d4b09b6bf8308928148e2197bfe","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-08-30T04:21:07Z","title_canon_sha256":"1665516e00ade9e98fc9ec05aa590ad60f883036dca1d5258cb12b831e950a52"},"schema_version":"1.0","source":{"id":"1409.0081","kind":"arxiv","version":1}},"canonical_sha256":"da70702359b86fbd86d90aac3a4f65bf67164aaa1eda8b5b2d70011e71866a5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da70702359b86fbd86d90aac3a4f65bf67164aaa1eda8b5b2d70011e71866a5a","first_computed_at":"2026-05-18T02:43:53.421641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:43:53.421641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hm5OA0ufr5VlEm1YMURjsFxNCEnS9ERSLFOULbQTaCxLC0Eqc4xJzVbadKFw+ErTRkdLDY03WxvtnDOjpy5WBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:43:53.422005Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.0081","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:617778144ee5f8be73e7a9984228257e7bdd4f2f48cfe25f2b32270535d1010d","sha256:8eb7158a96a1e69c55865f586e0c8ac180c433b52201691644c885e704711658"],"state_sha256":"1dd3acfd22ddb04cb87568030a20ae64336315a8c5318867ed3b55e7a143cf70"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gFCJIaNnwBI+D1Gbo39JDP5kap0pdFgN2ni6x4NgiZEtdU4g6JEIdYch7spIVlieDa7yVG7SqQHlbdpM4nw3Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T17:20:30.187194Z","bundle_sha256":"8710f54ce4954cbe816825e0d3a14013146519d9d8d161331d5f73aa1b76e9e7"}}