Decompositions of edge-coloured infinite complete graphs into monochromatic paths II

TL;DR

This paper proves that in any finite-edge coloured infinite complete graph, vertices can be partitioned into disjoint monochromatic paths of different colours, extending finite results to infinite graphs and answering a longstanding question.

Contribution

It extends finite-edge coloured graph decomposition results to infinite graphs, confirming R. Rado's 1978 question about monochromatic path partitions.

Findings

Vertices of infinite complete graphs can be partitioned into monochromatic paths of different colours.

The result generalizes finite graph decompositions to infinite settings.

Answers a 1978 open problem by R. Rado.

Abstract

P. Erd\H{o}s proved that every 2-edge coloured complete graph on the natural numbers can be vertex decomposed into two monochromatic paths of different colour. This result was extended by R. Rado to an arbitrary finite number of colours. We prove that the vertices of every finite-edge coloured infinite complete graph can be partitioned into disjoint monochromatic paths of different colours. This answers a question of R. Rado from 1978.