{"paper":{"title":"On the Topology of Weakly and Strongly Separated Set Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benjamin Hirsch, Daniel Hess","submitted_at":"2011-10-05T02:19:57Z","abstract_excerpt":"We examine the topology of the clique complexes of the graphs of weakly and strongly separated subsets of the set $[n]=\\{1,2,...,n\\}$, which, after deleting all cone points, we denote by $\\hat{\\Delta}_{ws}(n)$ and $\\hat{\\Delta}_{ss}(n)$, respectively. In particular, we find that $\\hat{\\Delta}_{ws}(n)$ is contractible for $n\\geq4$, while $\\hat{\\Delta}_{ss}(n)$ is homotopy equivalent to a sphere of dimension $n-3$. We also show that our homotopy equivalences are equivariant with respect to the group generated by two particular symmetries of $\\hat{\\Delta}_{ws}(n)$ and $\\hat{\\Delta}_{ss}(n)$: one "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.0880","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"}