On high discrepancy $1$-factorizations of complete graphs

We proved that for every sufficiently large $n$, the complete graph $K_{2n}$ with an arbitrary edge signing $\sigma :E(K_{2n})\to \{-1,+1\}$ admits a high discrepancy $1$-factor decomposition. That is, there exists a universal constant $c>0$ such that every edge-signed $K_{2n}$ has a perfect matching decomposition $\{{\psi }_1,\dots ,{\psi }_{2n-1}\}$, where for each perfect matching ${\psi }_i$, the discrepancy $\mid \frac{1}{n}{\sum }_{e\in E({\psi }_i)}\sigma (e)\mid$ is at least $c$.