In 1978, Richard Rado showed that every edge-coloured complete graph of countably infinite order can be partitioned into monochromatic paths of different colours. He asked whether this remains true for uncountable complete graphs and a notion of generalised paths. In 2016, Daniel Soukup answered this in the affirmative and conjectured that a similar result should hold for complete bipartite graphs with bipartition classes of the same infinite cardinality, namely that every such graph edge-coloured with r colours can be partitioned into 2 r — 1 monochromatic generalised paths with each colour being used at most twice. In the present paper, we give an affirmative answer to Soukup’s conjecture.

中文翻译：

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将边彩色无限完全二部图划分为单色路径

1978 年，理查德·拉多 (Richard Rado) 表明，每个可数无限阶的边着色完整图都可以划分为不同颜色的单色路径。他询问对于不可数完全图和广义路径的概念是否仍然如此。2016 年，Daniel Soukup 对此给出了肯定的回答，并推测类似的结果应该适用于具有相同无限基数的二分类的完整二分图，即每个这样的用 r 种颜色边着色的图都可以划分为 2 r - 1每种颜色最多使用两次的单色广义路径。在本文中，我们对 Soukup 的猜想给出了肯定的回答。