In an unfriendly coloring of a graph the color of every node mismatches that of the majority of its neighbors. We show that every probability measure preserving Borel graph with finite average degree admits a Borel unfriendly coloring almost everywhere. We also show that every bounded degree Borel graph of subexponential growth admits a Borel unfriendly coloring.

中文翻译：

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具有有限平均度的图的不友好着色

在不友好的图形着色中，每个节点的颜色与大多数邻居的颜色都不匹配。我们表明，每一个具有有限平均度的概率度量保持Borel图都几乎在所有地方都接受了Borel不友好的着色。我们还表明，每个次指数增长的有界Borel图都承认Borel不友好的着色。