{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:3C4W6U6VZWVARWRVXO2XQLOVAT","short_pith_number":"pith:3C4W6U6V","canonical_record":{"source":{"id":"1210.7043","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-26T02:36:35Z","cross_cats_sorted":["cs.CG","cs.DM"],"title_canon_sha256":"36e4cc35c1174db42cd2b145e218b0a02f8e856f0bc213c26c7a454b4d86e7fb","abstract_canon_sha256":"5d3b40625403b7c8f62e310a971bbf4e28d11bd2bea168bfbbbee4074c772a12"},"schema_version":"1.0"},"canonical_sha256":"d8b96f53d5cdaa08da35bbb5782dd504ecf3f9427fbf791d1908ac98c6c28bb3","source":{"kind":"arxiv","id":"1210.7043","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.7043","created_at":"2026-05-18T03:42:21Z"},{"alias_kind":"arxiv_version","alias_value":"1210.7043v1","created_at":"2026-05-18T03:42:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7043","created_at":"2026-05-18T03:42:21Z"},{"alias_kind":"pith_short_12","alias_value":"3C4W6U6VZWVA","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"3C4W6U6VZWVARWRV","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"3C4W6U6V","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:3C4W6U6VZWVARWRVXO2XQLOVAT","target":"record","payload":{"canonical_record":{"source":{"id":"1210.7043","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-26T02:36:35Z","cross_cats_sorted":["cs.CG","cs.DM"],"title_canon_sha256":"36e4cc35c1174db42cd2b145e218b0a02f8e856f0bc213c26c7a454b4d86e7fb","abstract_canon_sha256":"5d3b40625403b7c8f62e310a971bbf4e28d11bd2bea168bfbbbee4074c772a12"},"schema_version":"1.0"},"canonical_sha256":"d8b96f53d5cdaa08da35bbb5782dd504ecf3f9427fbf791d1908ac98c6c28bb3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:21.887697Z","signature_b64":"th1atNIICNmdCnKFvZo5nladmJ6wffXzp3ZjA4N5i5Z4Kdo2muhrnP7WWiDCbyOF1Tuvmi9GMVMmABVB8yF5BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8b96f53d5cdaa08da35bbb5782dd504ecf3f9427fbf791d1908ac98c6c28bb3","last_reissued_at":"2026-05-18T03:42:21.886989Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:21.886989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.7043","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":"2026-05-18T03:42:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wXMnuPdNEXxvqlMi3t2R3d2BlF/UcRxYfVtVEHrkQmyU5TN0J5rO4Hx+zijAqBWQLwjauEHBQYZl5yVxggyWBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:45:04.560655Z"},"content_sha256":"0e1e7a71f54e95e71d2dff4d00ab8e8b82998b466d634278faba2e7923d44236","schema_version":"1.0","event_id":"sha256:0e1e7a71f54e95e71d2dff4d00ab8e8b82998b466d634278faba2e7923d44236"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:3C4W6U6VZWVARWRVXO2XQLOVAT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Empty Monochromatic Simplices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM"],"primary_cat":"math.CO","authors_text":"Clemens Huemer, Jorge Urrutia, Oswin Aichholzer, Ruy Fabila-Monroy, Thomas Hackl","submitted_at":"2012-10-26T02:36:35Z","abstract_excerpt":"Let $S$ be a $k$-colored (finite) set of $n$ points in $\\mathbb{R}^d$, $d\\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \\leq k \\leq d$ we provide a lower bound of $\\Omega(n^{d-k+1+2^{-d}})$ and strengthen this to $\\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7043","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":"2026-05-18T03:42:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9kCL2oagjcm+ik84H0DWugTCVLaLikhjDvACp1HH+c4pvK8H0hFED96P4uwNMGPJIZlCqwQUPeq1CMjTI00ECw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:45:04.561003Z"},"content_sha256":"960c54a79f6de797bfa9c8d3106ffada62ac42327a45acc173276b77656c7df3","schema_version":"1.0","event_id":"sha256:960c54a79f6de797bfa9c8d3106ffada62ac42327a45acc173276b77656c7df3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3C4W6U6VZWVARWRVXO2XQLOVAT/bundle.json","state_url":"https://pith.science/pith/3C4W6U6VZWVARWRVXO2XQLOVAT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3C4W6U6VZWVARWRVXO2XQLOVAT/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":"2026-06-02T15:45:04Z","links":{"resolver":"https://pith.science/pith/3C4W6U6VZWVARWRVXO2XQLOVAT","bundle":"https://pith.science/pith/3C4W6U6VZWVARWRVXO2XQLOVAT/bundle.json","state":"https://pith.science/pith/3C4W6U6VZWVARWRVXO2XQLOVAT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3C4W6U6VZWVARWRVXO2XQLOVAT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:3C4W6U6VZWVARWRVXO2XQLOVAT","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":"5d3b40625403b7c8f62e310a971bbf4e28d11bd2bea168bfbbbee4074c772a12","cross_cats_sorted":["cs.CG","cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-26T02:36:35Z","title_canon_sha256":"36e4cc35c1174db42cd2b145e218b0a02f8e856f0bc213c26c7a454b4d86e7fb"},"schema_version":"1.0","source":{"id":"1210.7043","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.7043","created_at":"2026-05-18T03:42:21Z"},{"alias_kind":"arxiv_version","alias_value":"1210.7043v1","created_at":"2026-05-18T03:42:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7043","created_at":"2026-05-18T03:42:21Z"},{"alias_kind":"pith_short_12","alias_value":"3C4W6U6VZWVA","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"3C4W6U6VZWVARWRV","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"3C4W6U6V","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:960c54a79f6de797bfa9c8d3106ffada62ac42327a45acc173276b77656c7df3","target":"graph","created_at":"2026-05-18T03:42:21Z","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 $S$ be a $k$-colored (finite) set of $n$ points in $\\mathbb{R}^d$, $d\\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \\leq k \\leq d$ we provide a lower bound of $\\Omega(n^{d-k+1+2^{-d}})$ and strengthen this to $\\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\\math","authors_text":"Clemens Huemer, Jorge Urrutia, Oswin Aichholzer, Ruy Fabila-Monroy, Thomas Hackl","cross_cats":["cs.CG","cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-26T02:36:35Z","title":"Empty Monochromatic Simplices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7043","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":"sha256:0e1e7a71f54e95e71d2dff4d00ab8e8b82998b466d634278faba2e7923d44236","target":"record","created_at":"2026-05-18T03:42:21Z","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":"5d3b40625403b7c8f62e310a971bbf4e28d11bd2bea168bfbbbee4074c772a12","cross_cats_sorted":["cs.CG","cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-26T02:36:35Z","title_canon_sha256":"36e4cc35c1174db42cd2b145e218b0a02f8e856f0bc213c26c7a454b4d86e7fb"},"schema_version":"1.0","source":{"id":"1210.7043","kind":"arxiv","version":1}},"canonical_sha256":"d8b96f53d5cdaa08da35bbb5782dd504ecf3f9427fbf791d1908ac98c6c28bb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8b96f53d5cdaa08da35bbb5782dd504ecf3f9427fbf791d1908ac98c6c28bb3","first_computed_at":"2026-05-18T03:42:21.886989Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:21.886989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"th1atNIICNmdCnKFvZo5nladmJ6wffXzp3ZjA4N5i5Z4Kdo2muhrnP7WWiDCbyOF1Tuvmi9GMVMmABVB8yF5BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:21.887697Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.7043","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e1e7a71f54e95e71d2dff4d00ab8e8b82998b466d634278faba2e7923d44236","sha256:960c54a79f6de797bfa9c8d3106ffada62ac42327a45acc173276b77656c7df3"],"state_sha256":"c57281763afb3ae20e778e5d24b242ea6754635c236bbf18c07822401b4b1655"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jmoPNmuGHlNExohSYyM3bgcJFCDkPsZt8n1mZAWH3irg1shPohtB8snCBmMHHxGXtrZAFfcU8DUUSgmNzh0DBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T15:45:04.563020Z","bundle_sha256":"233eeaf6c99aa421f3e2b45b68af422934ddfd1c96cf0f8e73cf95a152c77c94"}}