Cardinality of order intervals in linear lattices and of their sets of extreme points

We characterize pairs $\mathfrak {n}$, $\mathfrak {m}$ of cardinals with the property that there exist an Archimedean linear lattice $X$ and an order interval in $X$ such that $\mathfrak {n}$ is its cardinality while $\mathfrak {m}$ is the cardinality of the set of its extreme points. We also present analogous results, complete or partial, in the case where $X$ is additionally required to be nonatomic, atomic, Dedekind $\sigma $-complete, hyper-Archimedean, or to be a $C(K)$-space, where $K$ is a compact space.