Decompositions of edge-colored infinite complete graphs into monochromatic paths

• Published in: Discrete mathematics Vol. 340; no. 8; pp. 2053 - 2069
• Main Authors: Elekes, Márton, Soukup, Dániel T., Soukup, Lajos, Szentmiklóssy, Zoltán
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2017
• Subjects: Complete hypergraph; Edge coloring; Graph partition; Infinite complete graph; Monochromatic path; Path square; Uncountable complete graph; Graph partition; Complete hypergraph; Path square; Infinite complete graph; Edge coloring; Monochromatic path; Uncountable complete graph
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2016.09.028

An r-edge coloring of a graph or hypergraph G=(V,E) is a map c:E→{0,…,r−1}. Extending results of Rado and answering questions of Rado, Gyárfás and Sárközy we prove that •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);•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 kth powers of paths apart from a finite set (a kth power of a path is a sequence v0,v1,… of distinct vertices such that 1⩽|i−j|⩽k implies that vivj is an edge);•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;•the vertex set of every 2-edge colored complete graph on ω1 can be partitioned into 2 monochromatic paths with distinct colors.