So there are three sequences of integers denoted $s_1, s_2$ and $s_3$. Each of them starts at $2$ and goes up to an integer $N$. In other words, they're all identical sequences of numbers of the form $\{2,\dots,N\}$.
The question is as such:
Is there a way to make triplets of numbers $\{a,b,c\} \forall (a \in s_1, b \in s_2,c \in s_3) $ such that every number in a triplet is coprime to the other two and no number of any of the three sequences is left out of a triplet, or show up in more than one of them?
With the sequences starting at $2$, no. There are too many even numbers around. Two of them will wind up in the same triplet. If you start at $1$ you can succeed with $N=1,3$ but not otherwise.