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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)"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1708.09777","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-31T15:32:26Z","cross_cats_sorted":[],"title_canon_sha256":"c2d7dec2469e73ac6e2965b3325232191a5416ca1c3f9680f4fd639258996c8d","abstract_canon_sha256":"98a0eb1ddd4871a8eb1e0a4c5238adc2dd8c01a3c846e4373156e0208cb8b044"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:15.115925Z","signature_b64":"SKQYhPnfJZZkQvK4bIKZX7Bmz1D9aCs9muB/spGod2taCikOowuQqZFyxnz2xQEC7UFb52lipSIS8dZkzNxeAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8dc49f126b10aac2e56f822420f7c1032bfd4fd4658f43226042382a7c02ac42","last_reissued_at":"2026-05-18T00:36:15.115203Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:15.115203Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $K_n$, $\\sum_{e\\in E(H)}f(e)\\neq 0$. 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