Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a following quantitative statement:

Is there a finite number of triples in $\mathbb{Z}^{3}$ such that if the vectors $v_1,v_2,v_3$ are linearly dependent, then for some triple $(k_1,k_2,k_3)\neq 0$ we have $$k_{1}v_{1}+k_{2}v_{2}+k_{3}v_{3}=0?$$

The important thing is that the collection should work for all triples of linearly dependent vectors under consideration. If the answer to this question is NO, can one then give a infinite collection of such "test" integer triples that would have a very small density in $\mathbb{Z}^{3}$? That is, we clearly do not need to consider all triples of integers - having considered (1,2,3), we do not need (2,4,6) in the collection etc.