{"paper":{"title":"The de Bruijn-Erd\\H{o}s theorem from a Hausdorff measure point of view","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Christos Pelekis, Martin Dole\\v{z}al, Themis Mitsis","submitted_at":"2018-05-28T15:39:32Z","abstract_excerpt":"Motivated by a well-known result in extremal set theory, due to Nicolaas Govert de Bruijn and Paul Erd\\H{o}s, we consider curves in the unit $n$-cube $[0,1]^n$ of the form \\[ A=\\{(x,f_1(x),\\ldots,f_{n-2}(x),\\alpha): x\\in [0,1]\\}, \\] where $\\alpha$ is a fixed real number in $[0,1]$ and $f_1,\\ldots,f_{n-2}$ are injective measurable functions from $[0,1]$ to $[0,1]$. We refer to such a curve $A$ as an $n$-\\emph{de~Bruijn-Erd\\H{o}s-set}. Under the additional assumption that all functions $f_i,i=1,\\ldots,n-2,$ are piecewise monotone, we show that the Hausdorff dimension of $A$ is at most $1$ as wel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10980","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"}