{"paper":{"title":"A Note on Weighted Rooted Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander York, Talon Ward, Zi-Xia Song","submitted_at":"2015-04-16T21:37:36Z","abstract_excerpt":"Let $T$ be a tree rooted at $r$. Two vertices of $T$ are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets $A$ and $B$ of $V(T)$ are unrelated if, for any $a\\in A$ and $b\\in B$, $a$ and $b$ are unrelated. Let $\\omega$ be a nonnegative weight function defined on $V(T)$ with $\\sum_{v\\in V(T)}\\omega(v)=1$. In this note, we prove that either there is an $(r, u)$-path $P$ with $\\sum_{v\\in V(P)}\\omega(v)\\ge \\frac13$ for some $u\\in V(T)$, or there exist unrelated sets $A, B\\subseteq V(T)$ such that $\\sum_{a\\in A }\\omega(a)\\ge \\frac13$ and $\\sum_{b\\in B }\\omega(b)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04392","kind":"arxiv","version":3},"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"}