  For each infinite cardinal $\kappa $ and each graph $G=(V,E)$, we say that a
partition $\pi :V\rightarrow \left\{ 0,1\right\} $ is $\kappa $-unfriendly if,
for each $x\in V$, $\left| \left\{ y\in V\mid \left\{ x,y\right\} \in E\text{
and }\pi (y)\neq \pi (x)\right\} \right| $ is $\geq \left| \left\{ y\in V\mid
\left\{ x,y\right\} \in E\text{ and }\pi (y)=\pi (x)\right\} \right| $ or $\geq
\kappa $; $\pi $ is unfriendly if the first property is true for each $x\in V$.
Some uncountable graphs of infinite minimum degree without unfriendly partition
have been constructed by S. Shelah and E.C. Milner, but it is not presently
known if countable graphs without unfriendly partition exist. We show that, for
each integer $n$, each graph of infinite minimum degree has an $\omega
_{n}$-unfriendly partition. We also prove that the following properties are
equivalent: (i) each graph has an $\omega $-unfriendly partition; (ii) each
countable graph has an unfriendly partition; (iii) each countable graph without
nonempty induced subgraph of infinite minimum degree has an unfriendly
partition (actually it is enough to consider a smaller class of graphs).
