  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 $f:E(K_n)\to \{-1,1\}$
such that $|\sum_{e\in E(K_n)}f(e)|\leq \binom{n}{2}-h(n)$, where $h(n)=2(n+1)$
if $n \equiv 0$ (mod $4$) and $h(n)=2n$ if $n \not\equiv 0$ (mod $4$), there is
always a copy $H$ of $K_4$ in $K_n$ for which $\sum_{e\in E(H)}f(e)=0$, and the
value of $h(n)$ is sharp.
