Consider a 3-dimensional projective space $X$.
Let $m$ be the smallest number so that there are $m$ pairs of lines $ \ell_1,\ell'_1$, $ \ell_2,\ell_2'$, ... , $\ell_m, \ell'_m$ in $X$:
a) For every $ i=1,2,\dots,m$, $\ell_i \cap \ell'_i = \emptyset$.
b) For every $i,j \le m$, $i \ne j$, $\ell_i \cap \ell'_j \ne \emptyset$.
(If there is no upper bound on $m$ we let $m=\infty$.)
a) Is it always the case that $m=6$?
b) Is it (at least) true that either $m=6$ or $m=\infty$?
c) What is the answer for the projective space over the Quaternions?
1) Since $X$ has dimension greater than 2 it must be Desarguian and therefore there is a division ring $D$ the points and lines in $X$ corresponds to the 1-dimensional and 2-dimensional subspaces over $D^4$.
2) We can always construct 6 pairs by considering a standard basis for $D^4$, the six 2-dimensional spaces $V_1,\dots V_6$ spanned by pairs of basis elements and letting $U_i$ correspond to the complement pair of basis elements to the pair used for $V_i$. Therefore, $m \ge 6$.
3) If $X$ is Papussian, namely if $D$ is a field $F$, then there is an argument based on the exterior algebra that $m \le 6$. The intersection property implies that the vectors in $\bigwedge^2(F^4)$ which correspond to the 2-dimensional vectors represented by the $m$ lines are linearly independent. This is a special case of a theorem by Lovasz. For more details see this blog post.
A question of independent interest that might be related is:
Question d) Is there some analog of the exterior algebra over division rings?
4) Benjy Weiss brought to my attention a paper by S.A. Amitsur Rational identities and applications to algebra and geometry. In the paper it is shown that, for Desarguian geometries, intersection theorems are equivalent to rational identities in the coordinate ring and that any nontrivial intersection theorem, together with the order axioms, implies Pappus' theorem. This may be relevant for showing that always, or at least in some cases, if you cannot find $m$ pairs with $m>6$ then $ m=\infty$. However I cannot tell if Amitsur's theorem covers the case at hand.