{"paper":{"title":"The extremal functions of classes of matroids of bounded branch-width","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Rohan Kapadia","submitted_at":"2015-03-27T03:59:43Z","abstract_excerpt":"For a set of matroids $\\mathcal{M}$, let $ex_\\mathcal{M}(n)$ be the maximum size of a simple rank-$n$ matroid in $\\mathcal{M}$. We prove that, for any finite field $\\mathbb{F}$, if $\\mathcal{M}$ is a minor-closed class of $\\mathbb{F}$-representable matroids of bounded branch-width, then $\\lim_{n \\rightarrow \\infty} ex_\\mathcal{M}(n) / n$ exists and is a rational number, $\\Delta$. We also show that $ex_\\mathcal{M}(n) - \\Delta n$ is periodic when $n$ is sufficiently large and that $ex_\\mathcal{M}$ is achieved by a subclass of $\\mathcal{M}$ of bounded path-width."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07954","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"}