Given a finite irreducible Coxeter group $W$, what’s the largest subset $K\subseteq W$ such that for all $u,v \in K$, it is not true that
$u <_R v$ (nor $v <_R u$)
where $<_R$ denotes the weak right order?
It would be interesting to know specific examples of such maximal sets and whether a nice general answer exists (possibly for all Coxeter groups and the strong Bruhat order).
I wouldn’t be surprised if a general theorem for finding the largest incomparable subsets of suitably nice posets exists but I haven’t found it.