A majority coloring of an undirected graph is a vertex coloring in which for each vertex there are at least as many bi-chromatic edges containing that vertex as monochromatic ones. It is known that for every countable graph a majority 3-coloring always exists. The Unfriendly Partition Conjecture states that every countable graph admits a majority 2-coloring. Since the 3-coloring result extends to countable DAGs, a variant of the conjecture states that 2 colors are enough to majority color every countable DAG. We show that this is false by presenting a DAG for which 3 colors are necessary. Presented construction is strongly based on a StackExchange conversation regarding labellings of infinite graphs that is linked in the references.

翻译：无向图的多数着色是一种顶点着色，其中对于每个顶点，包含该顶点的双色边数至少不少于单色边数。已知对于每个可数图，总存在多数3-着色。非友好划分猜想声称每个可数图都存在多数2-着色。由于3-着色结果可推广至可数有向无环图，该猜想的变体声称2种颜色足以对每个可数有向无环图进行多数着色。我们通过构造一个需要3种颜色的有向无环图，证明该结论不成立。所提出的构造方法主要基于参考文献中链接的关于无限图标号的StackExchange讨论。