The closed cone of flag vectors of Eulerian partially ordered sets is studied. It is completely determined up through rank seven. HalfEulerian 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.

This set of notes is prepared for the Meander Group (MG) at Brigham Young University. Its purpose is to introduce MG to: (1) the basic definitions and theorems of partially ordered set theory and (2) the various combinatorial methods associated with partially ordered sets.

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly th...