{"paper":{"title":"Constructing large k-systems on Surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GT","authors_text":"Tarik Aougab","submitted_at":"2014-03-20T12:58:31Z","abstract_excerpt":"Let $S_{g}$ denote the genus $g$ closed orientable surface. For $k\\in \\mathbb{N}$, a $k$-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than $k$ times. Juvan-Malni\\v{c}-Mohar \\cite{Ju-Mal-Mo} showed that there exists a $k$-system on $S_{g}$ whose size is on the order of $g^{k/4}$. For each $k\\geq 2$, We construct a $k$-system on $S_{g}$ with on the order of $g^{\\lfloor (k+1)/2 \\rfloor +1}$ elements. The $k$-systems we construct behave well with respect to subsurface inclusion, analogously to how a pants decomposition contains pants decompo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.5123","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"}