Factorizing the Rado graph and infinite complete graphs

Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be a family of infinite graphs, together with $\Lambda$. The Factorization Problem $FP(\mathcal{F}, \Lambda)$ asks whether $\mathcal{F}$ can realized as factorization $\Lambda$, namely, there is $\mathcal{G}=\{\Gamma_{\alpha}: $\Lambda$ such that each $\Gamma_{\alpha}$ copy $F_{\alpha}$.We study this problem when either the Rado graph $R$ or complete $K_\aleph$ order $\aleph$. When countable family, we show R)$ solvable if and only in has no finite dominating set. We also prove K_\aleph)$ admits solution whenever cardinality coincides domination numbers its graphs.For some non existence results graphs are finite. More precisely, $K_\N$ into copies $k$-star (that is, vertex disjoint union $k$ stars) $k=1,2$, whereas it exists $k\geq 4$, leaving open for $k=3$.Finally, determine sufficient conditions decomposition to arranged resolution classes.