{"paper":{"title":"Pach's selection theorem does not admit a topological extension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eran Nevo, Imre B\\'ar\\'any, Martin Tancer, Roy Meshulam","submitted_at":"2016-10-17T11:37:53Z","abstract_excerpt":"Let $U_1,\\dots, U_{d+1}$ be $n$-element sets in $R^d$ and let $\\langle u_1,\\ldots,u_{d+1}\\rangle$ denote the convex hull of points $u_i$ in $U_i$ (for all $i$) which is a (possibly degenerate) simplex. Pach's selection theorem says that there are sets $Z_1 \\subset U_1,\\dots, Z_{d+1} \\subset U_{d+1}$ and a point $u$ in $R^d$ such that each $|Z_i| > c_1(d)n$ and $u$ belongs to $\\langle z_1,...,z_{d+1} \\rangle$ for every choice of $z_1$ in $Z_1,\\dots,z_{d+1}$ in $Z_{d+1}$. Here we show that this theorem does not admit a topological extension with linear size sets $Z_i$. However, there is a topolo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05053","kind":"arxiv","version":2},"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"}