Partitioning edge-coloured infinite complete bipartite graphs into monochromatic paths

TL;DR

This paper proves that uncountably infinite bipartite graphs with edges coloured in multiple colours can be partitioned into a bounded number of monochromatic generalized paths, confirming a conjecture by Soukup.

Contribution

It provides a proof confirming Soukup's conjecture that such bipartite graphs can be partitioned into 2r-1 monochromatic paths with each colour used at most twice.

Findings

Confirmed Soukup's conjecture for uncountable bipartite graphs.

Established bounds on the number of paths needed for partition.

Extended Rado's 1978 result to uncountable bipartite graphs.

Abstract

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 \emph{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 $2r-1$ monochromatic generalised paths with each colour being used at most twice. In the present paper, we give an affirmative answer to Soukup's conjecture.