{"paper":{"title":"Zero-sum $K_m$ over $\\mathbb{Z}$ and the story of $K_4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adriana Hansberg, Amanda Montejano, Yair Caro","submitted_at":"2017-08-31T15:32:26Z","abstract_excerpt":"We prove the following results solving a problem raised in [Y. Caro, R. Yuster, On zero-sum and almost zero-sum subgraphs over $\\mathbb{Z}$, Graphs Combin. 32 (2016), 49--63]. For a positive integer $m\\geq 2$, $m\\neq 4$, there are infinitely many values of $n$ such that the following holds: There is a weighting function $f:E(K_n)\\to \\{-1,1\\}$ (and hence a weighting function $f: E(K_n)\\to \\{-1,0,1\\}$), such that $\\sum_{e\\in E(K_n)}f(e)=0$ but, for every copy $H$ of $K_m$ in $K_n$, $\\sum_{e\\in E(H)}f(e)\\neq 0$. On the other hand, for every integer $n\\geq 5$ and every weighting function $f:E(K_n)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.09777","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"}