Given a finite number of primitive elements $v_1,\dots,v_k\in\mathbb{Z}^{n+1}$ (i.e. the gcd of the entries of each $v_i$ is $\pm1$), is it always possible to find an element $v\in\mathbb{Z}^{n+1}$ such that each of the sets $\{v_i,v\}$ can be extended to a basis of $\mathbb{Z}^{n+1}$? If not, what conditions do we need to impose on $k$, $n$ and the $v_i$?
For $n=1$ it is not always possible, a counterexample is given by $v_1=(1,2)$, $v_2=(3,1)$, but I am not sure about the case $n\geq2$. If we replace $\mathbb{Z}$ by $\mathbb{R}$ then it of course holds for all $n\in\mathbb{N}$ as we just need to find a line in $\mathbb{R}^{n+1}$ that does not contain any of the $v_i$.
More generally, assume we have a finite number of subsets $\{v_{1,1},\dots,v_{m,1}\},\dots,\{v_{1,k},\dots,v_{m,k}\}\subseteq\mathbb{Z}^{n+m}$ such that each subset can be extended to a basis of $\mathbb{Z}^{n+m}$. Is it always possible to find an element $v\in\mathbb{Z}^{n+m}$ such that each of the sets $\{v_{1,1},\dots,v_{m,1},v\},\dots,\{v_{1,k},\dots,v_{m,k},v\}$ can be extended to a basis of $\mathbb{Z}^{n+m}$?
