{"paper":{"title":"Topology of unavoidable complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AT","authors_text":"Du\\v{s}ko Joji\\'c, Rade T. \\v{Z}ivaljevi\\'c, Sini\\v{s}a T. Vre\\'cica, Wac{\\l}aw Marzantowicz","submitted_at":"2016-03-28T18:25:56Z","abstract_excerpt":"The partition number $\\pi(K)$ of a simplicial complex $K\\subset 2^{[m]}$ is the minimum integer $\\nu$ such that for each partition $A_1\\uplus\\ldots\\uplus A_\\nu = [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $\\pi(K)\\leq r$. We say that a complex $K$ is globally $r$-non-embeddable in $\\mathbb{R}^d$ if for each continuous map $f: | K| \\rightarrow \\mathbb{R}^d$ there exist $r$ vertex disjoint faces $\\sigma_1,\\ldots, \\sigma_r$ of $| K|$ such that $f(\\sigma_1)\\cap\\ldots\\cap f(\\sigma_r)\\neq\\emptyset$. Motivated by the problems of Tverberg-Van Kampen-Flor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08472","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"}