3 problems

Let $f:\binom{[n]}{2}\to [n]$ be the unique résumé of a diameter-$2$ path system $\mathcal{P}$. The system $\mathcal{P}$ is monotone if $f(i,j)\le f(i,k)$ whenever…

Let $G$ be a graph on $n$ vertices. A separating path system is a collection of paths in $G$ that separates the edges of $G$, either weakly or strongly. Linear-size separating path…

Asymptotic existence conjecture. Asymptotically almost every graph has an irreducible non-metrizable path system; specifically, this holds with probability