Unrestricted Red Size and Sign-Coherence

Abstract

The unrestricted red size of a quiver is the maximal number of red vertices in its framed quiver after any given mutation sequence. In a 2023 paper by E. Bucher and J. Machacek, it was shown that connected, mutation-finite quivers either have an unrestricted red size of $n-1$ or $n$, where $n$ is the number of vertices in the quiver. We prove here that the same holds for the connected, mutation-infinite case using forks. As such, the unrestricted red size for any quiver equals $n-c$, where $c$ is the number of connected components of the quiver that do not admit a reddening sequence. Additionally, we prove a result on the $c$-vectors of forks that allows us to show that the $c$-vectors of both abundant acyclic quivers on any number of vertices and mutation-cyclic quivers on three vertices are sign-coherent with only elementary methods.