{"paper":{"title":"Making spanning graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Humberto Naves, Julia B\\\"ottcher, Peter Allen, Yoshiharu Kohayakawa, Yury Person","submitted_at":"2017-11-14T21:00:32Z","abstract_excerpt":"We prove that for each $D\\ge 2$ there exists $c>0$ such that whenever $b\\le c\\big(\\tfrac{n}{\\log n}\\big)^{1/D}$, in the $(1:b)$ Maker-Breaker game played on $E(K_n)$, Maker has a strategy to guarantee claiming a graph $G$ containing copies of all graphs $H$ with $v(H)\\le n$ and $\\Delta(H)\\le D$. We show further that the graph $G$ guaranteed by this strategy also contains copies of any graph $H$ with bounded maximum degree and degeneracy at most $\\tfrac{D-1}{2}$. This lower bound on the threshold bias is sharp up to the $\\log$-factor when $H$ consists of $\\tfrac{n}{3}$ vertex-disjoint triangles"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05311","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"}