Decompositions of edge-colored infinite complete graphs into monochromatic paths

For r ∈ N\{0} 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 • every r-edge colored complete k-uniform hypergraph on N can be partitioned into r monochromatic tight paths with distinct colors (a tight path in a kuniform 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 every r-edge colored complete graph on N can be partitioned into M monochromatic k powers of paths apart from a finite set (a k power of a path is a sequence v0, v1, . . . of distinct vertices such that |i− j| ≤ k implies that {vi, vj} is an edge), • every 2-edge colored complete graph on N can be partitioned into 4 monochromatic squares of paths, but not necessarily into 3, • every 2-edge colored complete graph on ω1 can be partitioned into 2 monochromatic paths with distinct colors.