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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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