{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:YDQESCBETCAI7OLKQK5J4TTBVK","short_pith_number":"pith:YDQESCBE","canonical_record":{"source":{"id":"1509.03332","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-10T20:49:54Z","cross_cats_sorted":[],"title_canon_sha256":"5d2fe0a2868afc5bb3f0dc2fb29318729e7428c4cd49c786dcfacdd8811d7bb2","abstract_canon_sha256":"5fd6c39b3efa563c683197ba6d847dd7fac70b8e0a56131cb449c8f757e6f415"},"schema_version":"1.0"},"canonical_sha256":"c0e049082498808fb96a82ba9e4e61aa87805f6aec16b1a630cfa52166f0b28e","source":{"kind":"arxiv","id":"1509.03332","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.03332","created_at":"2026-05-18T01:33:23Z"},{"alias_kind":"arxiv_version","alias_value":"1509.03332v1","created_at":"2026-05-18T01:33:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03332","created_at":"2026-05-18T01:33:23Z"},{"alias_kind":"pith_short_12","alias_value":"YDQESCBETCAI","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"YDQESCBETCAI7OLK","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"YDQESCBE","created_at":"2026-05-18T12:29:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:YDQESCBETCAI7OLKQK5J4TTBVK","target":"record","payload":{"canonical_record":{"source":{"id":"1509.03332","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-10T20:49:54Z","cross_cats_sorted":[],"title_canon_sha256":"5d2fe0a2868afc5bb3f0dc2fb29318729e7428c4cd49c786dcfacdd8811d7bb2","abstract_canon_sha256":"5fd6c39b3efa563c683197ba6d847dd7fac70b8e0a56131cb449c8f757e6f415"},"schema_version":"1.0"},"canonical_sha256":"c0e049082498808fb96a82ba9e4e61aa87805f6aec16b1a630cfa52166f0b28e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:23.615673Z","signature_b64":"A7qibK5mXsJHTvtBIUm3gXCN8o7C1svUM2NUcsSADB+J8W0Tvg8KCnTSLUFqnCG5mUDIV9Ekr7iWK/CWRJHjAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0e049082498808fb96a82ba9e4e61aa87805f6aec16b1a630cfa52166f0b28e","last_reissued_at":"2026-05-18T01:33:23.615053Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:23.615053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.03332","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":"2026-05-18T01:33:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PkcTH5IRj+GKz6efhc5dtmO2IGapFRBVV/JasSiTpu4g6c0a8VEk9OPjhgkYb1/+GOsaPV4mD+txWtg59gTvBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T14:47:12.171838Z"},"content_sha256":"4efb3aa52231e822b9b3cdff8632143d8aae2afd00a480e30febea589bc910b2","schema_version":"1.0","event_id":"sha256:4efb3aa52231e822b9b3cdff8632143d8aae2afd00a480e30febea589bc910b2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:YDQESCBETCAI7OLKQK5J4TTBVK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Erd\\H{o}s-Szekeres without induction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sergey Norin, Yelena Yuditsky","submitted_at":"2015-09-10T20:49:54Z","abstract_excerpt":"Let $ES(n)$ be the minimal integer such that any set of $ES(n)$ points in the plane in general position contains $n$ points in convex position. The problem of estimating $ES(n)$ was first formulated by Erd\\H{o}s and Szekeres, who proved that $ES(n) \\leq \\binom{2n-4}{n-2}+1$. The current best upper bound, $\\lim\\sup_{n \\to \\infty} \\frac{ES(n)}{\\binom{2n-5}{n-2}}\\le \\frac{29}{32}$, is due to Vlachos. We improve this to $$\\lim\\sup_{n \\to \\infty} \\frac{ES(n)}{\\binom{2n-5}{n-2}}\\le \\frac{7}{8}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03332","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":"2026-05-18T01:33:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Tvr8aJH1PcHtx7GXqs+MRMsosTw86TD1Gn5Zsu7IWQSQyZVtP/4xiZMN8inWcTANX22HlB8P1Y9v/VBdppfQCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T14:47:12.172191Z"},"content_sha256":"fde6945a57ec9484e0ac838f72aa3d24164a5c46976ebe34eee6a009ad88fd4a","schema_version":"1.0","event_id":"sha256:fde6945a57ec9484e0ac838f72aa3d24164a5c46976ebe34eee6a009ad88fd4a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YDQESCBETCAI7OLKQK5J4TTBVK/bundle.json","state_url":"https://pith.science/pith/YDQESCBETCAI7OLKQK5J4TTBVK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YDQESCBETCAI7OLKQK5J4TTBVK/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":"2026-05-30T14:47:12Z","links":{"resolver":"https://pith.science/pith/YDQESCBETCAI7OLKQK5J4TTBVK","bundle":"https://pith.science/pith/YDQESCBETCAI7OLKQK5J4TTBVK/bundle.json","state":"https://pith.science/pith/YDQESCBETCAI7OLKQK5J4TTBVK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YDQESCBETCAI7OLKQK5J4TTBVK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:YDQESCBETCAI7OLKQK5J4TTBVK","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":"5fd6c39b3efa563c683197ba6d847dd7fac70b8e0a56131cb449c8f757e6f415","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-10T20:49:54Z","title_canon_sha256":"5d2fe0a2868afc5bb3f0dc2fb29318729e7428c4cd49c786dcfacdd8811d7bb2"},"schema_version":"1.0","source":{"id":"1509.03332","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.03332","created_at":"2026-05-18T01:33:23Z"},{"alias_kind":"arxiv_version","alias_value":"1509.03332v1","created_at":"2026-05-18T01:33:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03332","created_at":"2026-05-18T01:33:23Z"},{"alias_kind":"pith_short_12","alias_value":"YDQESCBETCAI","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"YDQESCBETCAI7OLK","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"YDQESCBE","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:fde6945a57ec9484e0ac838f72aa3d24164a5c46976ebe34eee6a009ad88fd4a","target":"graph","created_at":"2026-05-18T01:33:23Z","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":"Let $ES(n)$ be the minimal integer such that any set of $ES(n)$ points in the plane in general position contains $n$ points in convex position. The problem of estimating $ES(n)$ was first formulated by Erd\\H{o}s and Szekeres, who proved that $ES(n) \\leq \\binom{2n-4}{n-2}+1$. The current best upper bound, $\\lim\\sup_{n \\to \\infty} \\frac{ES(n)}{\\binom{2n-5}{n-2}}\\le \\frac{29}{32}$, is due to Vlachos. We improve this to $$\\lim\\sup_{n \\to \\infty} \\frac{ES(n)}{\\binom{2n-5}{n-2}}\\le \\frac{7}{8}.$$","authors_text":"Sergey Norin, Yelena Yuditsky","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-10T20:49:54Z","title":"Erd\\H{o}s-Szekeres without induction"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03332","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":"sha256:4efb3aa52231e822b9b3cdff8632143d8aae2afd00a480e30febea589bc910b2","target":"record","created_at":"2026-05-18T01:33:23Z","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":"5fd6c39b3efa563c683197ba6d847dd7fac70b8e0a56131cb449c8f757e6f415","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-10T20:49:54Z","title_canon_sha256":"5d2fe0a2868afc5bb3f0dc2fb29318729e7428c4cd49c786dcfacdd8811d7bb2"},"schema_version":"1.0","source":{"id":"1509.03332","kind":"arxiv","version":1}},"canonical_sha256":"c0e049082498808fb96a82ba9e4e61aa87805f6aec16b1a630cfa52166f0b28e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c0e049082498808fb96a82ba9e4e61aa87805f6aec16b1a630cfa52166f0b28e","first_computed_at":"2026-05-18T01:33:23.615053Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:23.615053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A7qibK5mXsJHTvtBIUm3gXCN8o7C1svUM2NUcsSADB+J8W0Tvg8KCnTSLUFqnCG5mUDIV9Ekr7iWK/CWRJHjAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:23.615673Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.03332","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4efb3aa52231e822b9b3cdff8632143d8aae2afd00a480e30febea589bc910b2","sha256:fde6945a57ec9484e0ac838f72aa3d24164a5c46976ebe34eee6a009ad88fd4a"],"state_sha256":"e79993390f536c61e71681eeb16b93ef9c02c1d183e87fb1a78f05d1f4a6dc29"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7uwGqxacloq3w+fLsSePYTw3beYG6QtvBGM5kqAWnqmeP9Yjd94871w5qC0/sP4iCARNjVfpc5L4de7/84d3BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T14:47:12.174518Z","bundle_sha256":"f4ffd039396f08cab501a2cc491f61cd264d5b2073e5a21a2aa654b77ee7fa0f"}}