  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 Cowen and Emerson that every graph has an unfriendly partition,
but Milner and Shelah found counterexamples for that statement by analyzing
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 that property
that admits no unfriendly partitions. Actually, among some other independence
results, we conclude that this size cannot be determined from the usual axioms
of set theory.
