{"paper":{"title":"Dense binary $PG(t-1,2)$-free matroids have critical number $t-1$ or $t$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jonathan Tidor","submitted_at":"2015-08-28T16:59:36Z","abstract_excerpt":"The critical threshold of a (simple binary) matroid $N$ is the infimum over all $\\rho$ such that any $N$-free matroid $M$ with $|M|>\\rho2^{r(M)}$ has bounded critical number. In this paper, we resolve two conjectures of Geelen and Nelson, showing that the critical threshold of the projective geometry $PG(t-1,2)$ is $1-3\\cdot2^{-t}$. We do so by proving the following stronger statement: if $M$ is $PG(t-1,2)$-free with $|M|>(1-3\\cdot2^{-t})2^{r(M)}$, then the critical number of $M$ is $t-1$ or $t$. Together with earlier results of Geelen and Nelson [GN14] and Govaerts and Storme [GS06], this com"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07278","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"}