2 problems

Mohr–Pardey–Rautenbach's conjecture. Fix integers $k,n\geq 1$ such that $\binom{n}{2}$ and $(k-1)n/k$ are both even integers. If $\mathbf{c}:E(K_n)\to\{-1,1\}$ is a 0-sum labeling…

Let $T$ be a tree with $t$ vertices, let $\varepsilon>0$, and let $\mathcal{P}(\varepsilon)$ be the property that all but an $O(\varepsilon)$-fraction of the edges of a graph can b…