{"paper":{"title":"Lower bounds on geometric Ramsey functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Edgardo Rold\\'an-Pensado, Ji\\v{r}\\'i Matou\\v{s}ek, Marek Eli\\'a\\v{s}, Zuzana Safernov\\'a","submitted_at":"2013-07-19T08:29:02Z","abstract_excerpt":"We continue a sequence of recent works studying Ramsey functions for semialgebraic predicates in $\\mathbb{R}^d$. A $k$-ary semialgebraic predicate $\\Phi(x_1,\\ldots,x_k)$ on $\\mathbb{R}^d$ is a Boolean combination of polynomial equations and inequalities in the $kd$ coordinates of $k$ points $x_1,\\ldots,x_k\\in\\mathbb{R}^d$. A sequence $P=(p_1,\\ldots,p_n)$ of points in $\\mathbb{R}^d$ is called $\\Phi$-homogeneous if either $\\Phi(p_{i_1}, \\ldots,p_{i_k})$ holds for all choices $1\\le i_1 < \\cdots < i_k\\le n$, or it holds for no such choice. The Ramsey function $R_\\Phi(n)$ is the smallest $N$ such t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5157","kind":"arxiv","version":3},"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"}