  A partially ordered set is r-thick if every nonempty open interval contains
at least r elements. This paper studies the flag vectors of graded, r-thick
posets and shows the smallest convex cone containing them is isomorphic to the
cone of flag vectors of all graded posets. It also defines a k-analogue of the
Mobius function and k-Eulerian posets, which are 2k-thick. Several
characterizations of k-Eulerian posets are given. The generalized
Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A
new inequality is proved to be valid and sharp for rank 8 Eulerian posets.
