Flag Vectors of Eulerian Partially Ordered Sets

The closed cone of flag vectors of Eulerian partially ordered sets is studied. A new family of linear inequalities valid for Eulerian flag vectors is given. 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. Other extreme posets are formed from consideration of the cd -index. The cone of Eulerian flag vectors is completely determined up through rank seven.