Decompositions of edge-colored infinite complete graphs into monochromatic paths

TL;DR

This paper extends classical results on edge-colored infinite graphs and hypergraphs, proving new partitioning theorems into monochromatic paths and powers, addressing questions posed by Rado, Gyárfás, and Sárközy.

Contribution

It introduces novel decompositions of infinite complete graphs and hypergraphs into monochromatic paths and powers, generalizing and strengthening previous results.

Findings

Partition of hypergraph vertices into monochromatic tight paths

Decomposition of infinite graphs into monochromatic path powers

Partitioning infinite graphs into monochromatic paths with minimal sets

Abstract

An $r$-edge coloring of a graph or hypergraph $G=(V,E)$ is a map $c:E\to \{0, \dots, r-1\}$. Extending results of Rado and answering questions of Rado, Gy\'arf\'as and S\'ark\"ozy we prove that (1.) the vertex set of every $r$-edge colored countably infinite complete $k$-uniform hypergraph can be partitioned into $r$ monochromatic tight paths with distinct colors (a tight path in a $k$-uniform hypergraph is a sequence of distinct vertices such that every set of $k$ consecutive vertices forms an edge), (2.) for all natural numbers $r$ and $k$ there is a natural number $M$ such that the vertex set of every $r$-edge colored countably infinite complete graph can be partitioned into $M$ monochromatic $k^{th}$ powers of paths apart from a finite set (a $k^{th}$ power of a path is a sequence $v_0, v_1, \dots$ of distinct vertices such that $1\le|i-j| \le k$ implies that $v_iv_j$ is an edge), (3.) the vertex set of every $2$-edge colored countably infinite complete graph can be partitioned into $4$ monochromatic squares of paths, but not necessarily into $3$, (4.) the vertex set of every $2$-edge colored complete graph on $\omega_1$ can be partitioned into $2$ monochromatic paths with distinct colors.