{"paper":{"title":"General Position Subsets and Independent Hyperplanes in d-Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Csaba D. T\\'oth, David R. Wood, Jean Cardinal","submitted_at":"2014-10-14T10:13:12Z","abstract_excerpt":"Erd\\H{o}s asked what is the maximum number $\\alpha(n)$ such that every set of $n$ points in the plane with no four on a line contains $\\alpha(n)$ points in general position. We consider variants of this question for $d$-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed $d$:\n  - Every set $H$ of $n$ hyperplanes in $\\mathbb{R}^d$ contains a subset $S\\subseteq H$ of size at least $c \\left(n \\log n\\right)^{1/d}$, for some constant $c=c(d)>0$, such that no cell of the arrangement of $H$ is bounded by hyperplanes of $S$ only.\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3637","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"}