### A generalization of a theorem of Erdös on asymptotic basis of order $2$

Let $\mathcal{T}$ be a system of disjoint subsets of ${\mathbb{N}}^{*}$. In this paper we examine the existence of an increasing sequence of natural numbers, $A$, that is an asymptotic basis of all infinite elements ${T}_{j}$ of $\mathcal{T}$ simultaneously, satisfying certain conditions on the rate of growth of the number of representations ${\mathit{r}}_{\mathit{n}}\left(\mathit{A}\right);{\mathit{r}}_{\mathit{n}}\left(\mathit{A}\right):=\left|\left\{({\mathit{a}}_{\mathit{i}},{\mathit{a}}_{\mathit{j}}):{\mathit{a}}_{\mathit{i}}\<{\mathit{a}}_{\mathit{j}};{\mathit{a}}_{\mathit{i}},{\mathit{a}}_{\mathit{j}}\in \mathit{A};\mathit{n}={\mathit{a}}_{\mathit{i}}+{\mathit{a}}_{\mathit{j}}\right\}\right|$, for all sufficiently large $n\in {T}_{j}$ and $j\in {\mathbb{N}}^{*}$ A theorem of P. Erdös is generalized.