{"paper":{"title":"Structure theorem for U5-free tournaments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gaku Liu","submitted_at":"2012-08-02T04:37:19Z","abstract_excerpt":"Let $U_5$ be the tournament with vertices $v_1$, ..., $v_5$ such that $v_2 \\rightarrow v_1$, and $v_i \\rightarrow v_j$ if $j-i \\equiv 1$, $2 \\pmod{5}$ and ${i,j} \\neq {1,2}$. In this paper we describe the tournaments which do not have $U_5$ as a subtournament. Specifically, we show that if a tournament $G$ is \"prime\"---that is, if there is no subset $X \\subseteq V(G)$, $1 < |X| < |V(G)|$, such that for all $v \\in V(G) \\backslash X$, either $v \\rightarrow x$ for all $x \\in X$ or $x \\rightarrow v$ for all $x \\in X$---then $G$ is $U_5$-free if and only if either $G$ is a specific tournament $T_n$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.0398","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"}