{"paper":{"title":"A fractal perspective on optimal antichains and intersecting subsets of the unit $n$-cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christos Pelekis, Konrad Engel, Themis Mitsis","submitted_at":"2017-07-16T10:47:14Z","abstract_excerpt":"An \\emph{$n$-cube antichain} is a subset of the unit $n$-cube $[0,1]^n$ that does not contain two elements $\\mathbf{x}=(x_1, x_2,\\ldots, x_n)$ and $\\mathbf{y}=(y_1, y_2,\\ldots, y_n)$ satisfying $x_i\\le y_i$ for all $i\\in \\{1,\\ldots,n\\}$. Using a chain partition of an adequate finite poset we show that the Hausdorff dimension of an $n$-cube antichain is at most $n-1$.We conjecture that the $(n-1)$-dimensional Hausdorff measure of an $n$-cube antichain is at most $n$ times the Hausdorff measure of a facet of the unit $n$-cube and we verify this conjecture for $n=2$ as well as under the assumptio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04856","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"}