{"paper":{"title":"Logarithmically-small Minors and Topological Minors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Richard Montgomery","submitted_at":"2013-09-30T15:39:40Z","abstract_excerpt":"Mader proved that for every integer $t$ there is a smallest real number $c(t)$ such that any graph with average degree at least $c(t)$ must contain a $K_t$-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with $n$ vertices and average degree at least $c(t)+\\epsilon$ must contain a $K_t$-minor consisting of at most $C(\\epsilon,t)\\log n$ vertices. Shapira and Sudakov subsequently proved that such a graph contains a $K_t$-minor consisting of at most $C(\\epsilon,t)\\log n \\log\\log n$ vertices. Here we build on their method using graph expansion to remove the $\\log\\log n$ factor and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7886","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"}