Unfriendly partitions when avoiding vertices of finite degree

Abstract

Abstract An unfriendly partition of a graph $G = (V,E)$ is a function $c: V \to 2$ such that $|\{x\in N(v): c(x)\neq c(v)\}|\geq |\{x\in N(v): c(x)=c(v)\}|$ for every vertex $v\in V$, where $N(v)$ denotes its neighborhood. It was conjectured by Cowan and Emerson [2] that every graph has an unfriendly partition, but Milner and Shelah in [5] found counterexamples for that statement by analysing graphs with uncountably many vertices. Curiously, none of their graphs have vertices with finite degree. Therefore, as a natural direction to approach, in this paper we search for the least cardinality of a graph with this property and that admits no unfriendly partitions. Actually, among some other independence results, we conclude that this value cannot be precisely determined within $\textrm {ZFC}$, in the sense that it may vary from model to model of set theory.