{"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"}