{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:4ZFJV5HN4MXHIMLSVQ5H26HI3F","short_pith_number":"pith:4ZFJV5HN","schema_version":"1.0","canonical_sha256":"e64a9af4ede32e743172ac3a7d78e8d95aaac87876a9f0d55100cffcb713a973","source":{"kind":"arxiv","id":"1812.06496","version":1},"attestation_state":"computed","paper":{"title":"Projection inequalities for antichains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Reiher, Christos Pelekis, Konrad Engel, Themis Mitsis","submitted_at":"2018-12-16T16:18:34Z","abstract_excerpt":"A set $A \\subseteq {\\mathbb{R}}^n$ is called an antichain (resp. antichain) if it does not contain two distinct elements ${\\mathbf x}=(x_1,\\ldots, x_n)$ and ${\\mathbf y}=(y_1,\\ldots, y_n)$ satisfying $x_i\\le y_i$ (resp. $x_i < y_i$) for all $i\\in \\{1,\\ldots,n\\}$. We show that the Hausdorff dimension of a weak antichain $A$ in the $n$-dimensional unit cube $[0,1]^n$ is at most $n-1$ and that the $(n-1)$-dimensional Hausdorff measure of $A$ is at most $n$, which are the best possible bounds. This result is derived as a corollary of the following {\\it projection inequality}, which may be of indep"},"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":"1812.06496","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-16T16:18:34Z","cross_cats_sorted":[],"title_canon_sha256":"94beea9cd1ae10cc24253220127910ae6d7fdc4c58112c4684f2980ff7b51ebc","abstract_canon_sha256":"995a0f7ec4d25610d84781c8cd9a98eaba0edb51cc7408deaf3487744f6d5f1c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:00:08.277263Z","signature_b64":"hCFbTQGKG43KwrVrqdEXeXOy+Po/1M0teq69V25mmE2kaBlWEGkis5IDby5VdhTkMe7Z+p7dCAWbrkof4t6zAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e64a9af4ede32e743172ac3a7d78e8d95aaac87876a9f0d55100cffcb713a973","last_reissued_at":"2026-07-05T02:00:08.276900Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:00:08.276900Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Projection inequalities for antichains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Reiher, Christos Pelekis, Konrad Engel, Themis Mitsis","submitted_at":"2018-12-16T16:18:34Z","abstract_excerpt":"A set $A \\subseteq {\\mathbb{R}}^n$ is called an antichain (resp. antichain) if it does not contain two distinct elements ${\\mathbf x}=(x_1,\\ldots, x_n)$ and ${\\mathbf y}=(y_1,\\ldots, y_n)$ satisfying $x_i\\le y_i$ (resp. $x_i < y_i$) for all $i\\in \\{1,\\ldots,n\\}$. We show that the Hausdorff dimension of a weak antichain $A$ in the $n$-dimensional unit cube $[0,1]^n$ is at most $n-1$ and that the $(n-1)$-dimensional Hausdorff measure of $A$ is at most $n$, which are the best possible bounds. This result is derived as a corollary of the following {\\it projection inequality}, which may be of indep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.06496","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/1812.06496/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1812.06496","created_at":"2026-07-05T02:00:08.276962+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.06496v1","created_at":"2026-07-05T02:00:08.276962+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.06496","created_at":"2026-07-05T02:00:08.276962+00:00"},{"alias_kind":"pith_short_12","alias_value":"4ZFJV5HN4MXH","created_at":"2026-07-05T02:00:08.276962+00:00"},{"alias_kind":"pith_short_16","alias_value":"4ZFJV5HN4MXHIMLS","created_at":"2026-07-05T02:00:08.276962+00:00"},{"alias_kind":"pith_short_8","alias_value":"4ZFJV5HN","created_at":"2026-07-05T02:00:08.276962+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.04727","citing_title":"On $k$-antichains in the unit $n$-cube","ref_index":7,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4ZFJV5HN4MXHIMLSVQ5H26HI3F","json":"https://pith.science/pith/4ZFJV5HN4MXHIMLSVQ5H26HI3F.json","graph_json":"https://pith.science/api/pith-number/4ZFJV5HN4MXHIMLSVQ5H26HI3F/graph.json","events_json":"https://pith.science/api/pith-number/4ZFJV5HN4MXHIMLSVQ5H26HI3F/events.json","paper":"https://pith.science/paper/4ZFJV5HN"},"agent_actions":{"view_html":"https://pith.science/pith/4ZFJV5HN4MXHIMLSVQ5H26HI3F","download_json":"https://pith.science/pith/4ZFJV5HN4MXHIMLSVQ5H26HI3F.json","view_paper":"https://pith.science/paper/4ZFJV5HN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.06496&json=true","fetch_graph":"https://pith.science/api/pith-number/4ZFJV5HN4MXHIMLSVQ5H26HI3F/graph.json","fetch_events":"https://pith.science/api/pith-number/4ZFJV5HN4MXHIMLSVQ5H26HI3F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4ZFJV5HN4MXHIMLSVQ5H26HI3F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4ZFJV5HN4MXHIMLSVQ5H26HI3F/action/storage_attestation","attest_author":"https://pith.science/pith/4ZFJV5HN4MXHIMLSVQ5H26HI3F/action/author_attestation","sign_citation":"https://pith.science/pith/4ZFJV5HN4MXHIMLSVQ5H26HI3F/action/citation_signature","submit_replication":"https://pith.science/pith/4ZFJV5HN4MXHIMLSVQ5H26HI3F/action/replication_record"}},"created_at":"2026-07-05T02:00:08.276962+00:00","updated_at":"2026-07-05T02:00:08.276962+00:00"}