  The closed cone of flag vectors of Eulerian partially ordered sets is
studied. It is completely determined up through rank seven. Half-Eulerian
posets are defined. Certain limit posets of Billera and Hetyei are
half-Eulerian; they give rise to extreme rays of the cone for Eulerian posets.
A new family of linear inequalities valid for flag vectors of Eulerian posets
is given.
