Irreducible nonmetrizable path systems in graphs

A path system   P ${\mathscr{P}}$ in a graphG = ( V , E )$G=(V,E)$ is a collection of paths with a uniqueu v $uv$ path for every two verticesu , v ∈ V $u,v\in V$ . We say thatP ${\mathscr{P}}$ is consistent if for any pathP ∈ P $P\in {\mathscr{P}}$ , every subpath ofP $P$ is also inP ${\mathscr{P}}$ . It is metrizable if there exists a positive weight functionw : E → R > 0$w:E\to {{\mathbb{R}}}_{\gt 0}$ such thatP ${\mathscr{P}}$ is comprised ofw $w$ ‐shortest paths. We callP ${\mathscr{P}}$irreducible if there does not exist a partitionV = A ⊔ B $V=A\bigsqcup B$ such thatP ${\mathscr{P}}$ restricts to a path system on bothG [ A ]$G[A]$ andG [ B ]$G[B]$ . In this paper, we construct an infinite family of nonmetrizable irreducible consistent path systems on certain Paley graphs.