I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. I want natural statements, e.g. statements that have been studied in number theory or combinatorics for their own sake. The motivation comes from looking for true statements that are not provable in $\mathsf{I}\Delta_0(L)$ where $L$ contains arbitrary fast growing (computable) functions.