{"paper":{"title":"A continuous analogue of Erd\\H{o}s' $k$-Sperner theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Christos Pelekis, Themis Mitsis, V\\'aclav Vlas\\'ak","submitted_at":"2019-04-21T17:00:48Z","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 show that the $1$-dimensional Hausdorff measure of a chain in the unit $n$-cube is at most $n$, and that the bound is sharp. Given this result, we consider the problem of maximising the $n$-dimensional Lebesgue measure of a measurable set $A\\subset [0,1]^n$ subject to the constraint that it satisfies $\\mathcal{H}^1(A\\cap C) \\le \\kappa$ for all chains $C\\su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.09625","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"}