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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 pathswith distinct colors (a tight path in a $k$-uniform hypergraph is a sequence ofdistinct vertices such that every set of $k$ consecutive vertices forms anedge),  (2.) for all natural numbers $r$ and $k$ there is a natural number $M$ suchthat the vertex set of every $r$-edge colored countably infinite complete graphcan be partitioned into $M$ monochromatic $k^{th}$ powers of paths apart from afinite set (a $k^{th}$ power of a path is a sequence $v_0, v_1, \dots$ ofdistinct vertices such that $1\le|i-j| \le k$ implies that $v_iv_j$ is anedge),  (3.) the vertex set of every $2$-edge colored countably infinite completegraph can be partitioned into $4$ monochromatic squares of paths, but notnecessarily 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.

Decompositions of edge-colored infinite complete graphs into monochromatic paths