2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78861The 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.31 pages, LaTeXCombinatorics06A07text