{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:4AF6KGF2CX6RUMI7GO5DIXF4EF","short_pith_number":"pith:4AF6KGF2","canonical_record":{"source":{"id":"1908.04727","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-13T16:39:45Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"02a6ef58af41df9cd6d8970c8e31380771b6f72faf23459a9a9f7f17bda19594","abstract_canon_sha256":"cfea24d6fb8d386e5ca0981ff1c135ed5a9fb53ea1c082648d07fba6e391c774"},"schema_version":"1.0"},"canonical_sha256":"e00be518ba15fd1a311f33ba345cbc216c481dd17945e4c7d4f549997bea92e3","source":{"kind":"arxiv","id":"1908.04727","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.04727","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"arxiv_version","alias_value":"1908.04727v1","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.04727","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"pith_short_12","alias_value":"4AF6KGF2CX6R","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"pith_short_16","alias_value":"4AF6KGF2CX6RUMI7","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"pith_short_8","alias_value":"4AF6KGF2","created_at":"2026-07-04T23:55:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:4AF6KGF2CX6RUMI7GO5DIXF4EF","target":"record","payload":{"canonical_record":{"source":{"id":"1908.04727","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-13T16:39:45Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"02a6ef58af41df9cd6d8970c8e31380771b6f72faf23459a9a9f7f17bda19594","abstract_canon_sha256":"cfea24d6fb8d386e5ca0981ff1c135ed5a9fb53ea1c082648d07fba6e391c774"},"schema_version":"1.0"},"canonical_sha256":"e00be518ba15fd1a311f33ba345cbc216c481dd17945e4c7d4f549997bea92e3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:55:15.916116Z","signature_b64":"gBihcL58Qdhi9u5qyBmCiSd4Z7VEu9wMDD+fyULhDTM+pVDK+nzDBtsTvSfbt11QcuacJfKM+EC2098gyvj7DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e00be518ba15fd1a311f33ba345cbc216c481dd17945e4c7d4f549997bea92e3","last_reissued_at":"2026-07-04T23:55:15.915645Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:55:15.915645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.04727","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":"2026-07-04T23:55:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0iVQ4ST8eS3oGQCnVK4IuBzZwvLEpRAp1B9CZRgKYc0uWbPJSQsC2vXy3iMrFpky4uuU+0Ptpv4uV8QIr228Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T14:18:53.313629Z"},"content_sha256":"601b3f4bcad8096c68a3b01c4090199646d68f9e8f92df72e64cbee3ccf7ad27","schema_version":"1.0","event_id":"sha256:601b3f4bcad8096c68a3b01c4090199646d68f9e8f92df72e64cbee3ccf7ad27"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:4AF6KGF2CX6RUMI7GO5DIXF4EF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On $k$-antichains in the unit $n$-cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Christos Pelekis, V\\'aclav Vlas\\'ak","submitted_at":"2019-08-13T16:39:45Z","abstract_excerpt":"A \\emph{chain} in the unit $n$-cube is a set $C\\subset [0,1]^n$ such that for every $\\mathbf{x}=(x_1,\\ldots,x_n)$ and $\\mathbf{y}=(y_1,\\ldots,y_n)$ in $C$ we either have $x_i\\le y_i$ for all $i\\in [n]$, or $x_i\\ge y_i$ for all $i\\in [n]$. We consider subsets, $A$, of the unit $n$-cube $[0,1]^n$ that satisfy \\[ \\text{card}(A \\cap C) \\le k, \\, \\text{ for all chains } \\, C \\subset [0,1]^n \\, , \\] where $k$ is a fixed positive integer. We refer to such a set $A$ as a $k$-antichain. We show that the $(n-1)$-dimensional Hausdorff measure of a $k$-antichain in $[0,1]^n$ is at most $kn$ and that the b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04727","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/1908.04727/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:55:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dYh4uWp1TN/mYTRnMjvUm9cnwIGmgHy0NygsK8tnMeJau3GwzF1Is9RKAmS+KerKSO6M+Y2teXllAiQrMBiHDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T14:18:53.314140Z"},"content_sha256":"e4827537a2476ce9c382109650c227d95b2772b4dd59f8d0093e3a1a2b3bd7db","schema_version":"1.0","event_id":"sha256:e4827537a2476ce9c382109650c227d95b2772b4dd59f8d0093e3a1a2b3bd7db"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4AF6KGF2CX6RUMI7GO5DIXF4EF/bundle.json","state_url":"https://pith.science/pith/4AF6KGF2CX6RUMI7GO5DIXF4EF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4AF6KGF2CX6RUMI7GO5DIXF4EF/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":"2026-08-16T14:18:53Z","links":{"resolver":"https://pith.science/pith/4AF6KGF2CX6RUMI7GO5DIXF4EF","bundle":"https://pith.science/pith/4AF6KGF2CX6RUMI7GO5DIXF4EF/bundle.json","state":"https://pith.science/pith/4AF6KGF2CX6RUMI7GO5DIXF4EF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4AF6KGF2CX6RUMI7GO5DIXF4EF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:4AF6KGF2CX6RUMI7GO5DIXF4EF","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":"cfea24d6fb8d386e5ca0981ff1c135ed5a9fb53ea1c082648d07fba6e391c774","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-13T16:39:45Z","title_canon_sha256":"02a6ef58af41df9cd6d8970c8e31380771b6f72faf23459a9a9f7f17bda19594"},"schema_version":"1.0","source":{"id":"1908.04727","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.04727","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"arxiv_version","alias_value":"1908.04727v1","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.04727","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"pith_short_12","alias_value":"4AF6KGF2CX6R","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"pith_short_16","alias_value":"4AF6KGF2CX6RUMI7","created_at":"2026-07-04T23:55:15Z"},{"alias_kind":"pith_short_8","alias_value":"4AF6KGF2","created_at":"2026-07-04T23:55:15Z"}],"graph_snapshots":[{"event_id":"sha256:e4827537a2476ce9c382109650c227d95b2772b4dd59f8d0093e3a1a2b3bd7db","target":"graph","created_at":"2026-07-04T23:55:15Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.04727/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A \\emph{chain} in the unit $n$-cube is a set $C\\subset [0,1]^n$ such that for every $\\mathbf{x}=(x_1,\\ldots,x_n)$ and $\\mathbf{y}=(y_1,\\ldots,y_n)$ in $C$ we either have $x_i\\le y_i$ for all $i\\in [n]$, or $x_i\\ge y_i$ for all $i\\in [n]$. We consider subsets, $A$, of the unit $n$-cube $[0,1]^n$ that satisfy \\[ \\text{card}(A \\cap C) \\le k, \\, \\text{ for all chains } \\, C \\subset [0,1]^n \\, , \\] where $k$ is a fixed positive integer. We refer to such a set $A$ as a $k$-antichain. We show that the $(n-1)$-dimensional Hausdorff measure of a $k$-antichain in $[0,1]^n$ is at most $kn$ and that the b","authors_text":"Christos Pelekis, V\\'aclav Vlas\\'ak","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-13T16:39:45Z","title":"On $k$-antichains in the unit $n$-cube"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04727","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":"sha256:601b3f4bcad8096c68a3b01c4090199646d68f9e8f92df72e64cbee3ccf7ad27","target":"record","created_at":"2026-07-04T23:55:15Z","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":"cfea24d6fb8d386e5ca0981ff1c135ed5a9fb53ea1c082648d07fba6e391c774","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-13T16:39:45Z","title_canon_sha256":"02a6ef58af41df9cd6d8970c8e31380771b6f72faf23459a9a9f7f17bda19594"},"schema_version":"1.0","source":{"id":"1908.04727","kind":"arxiv","version":1}},"canonical_sha256":"e00be518ba15fd1a311f33ba345cbc216c481dd17945e4c7d4f549997bea92e3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e00be518ba15fd1a311f33ba345cbc216c481dd17945e4c7d4f549997bea92e3","first_computed_at":"2026-07-04T23:55:15.915645Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:55:15.915645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gBihcL58Qdhi9u5qyBmCiSd4Z7VEu9wMDD+fyULhDTM+pVDK+nzDBtsTvSfbt11QcuacJfKM+EC2098gyvj7DQ==","signature_status":"signed_v1","signed_at":"2026-07-04T23:55:15.916116Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.04727","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:601b3f4bcad8096c68a3b01c4090199646d68f9e8f92df72e64cbee3ccf7ad27","sha256:e4827537a2476ce9c382109650c227d95b2772b4dd59f8d0093e3a1a2b3bd7db"],"state_sha256":"a7b5e771636903b647b4f8860c57ff0ddb0fa0fa576f5bd3be3e9afe8866ae57"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wfj6Z+s95UtoCOJfnCZrgsxO0p1yqHdb8BrpzaXc5tap5Ku7QgNqoA7vK/LTAiGmjxQ/lJToJlE3kizoW1tYDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T14:18:53.326229Z","bundle_sha256":"458f064ffd3d21ef451e5a30bc43cf61b37674d0fa3889942b873b7648ef578f"}}