  The combinatorial Hopf algebra on building sets $BSet$ extends the chromatic
Hopf algebra of simple graphs. The image of a building set under canonical
morphism to quasi-symmetric functions is the chromatic symmetric function of
the corresponding hypergraph. By passing from graphs to building sets, we
construct a sequence of symmetric functions associated to a graph. From the
generalized Dehn-Sommerville relations for the Hopf algebra $BSet$, we define a
class of building sets called eulerian and show that eulerian building sets
satisfy Bayer-Billera relations. We show the existence of the
$\mathbf{c}\mathbf{d}-$index, the polynomial in two noncommutative variables
associated to an eulerian building set. The complete characterization of
eulerian building sets is given in terms of combinatorics of intersection
posets of antichains of finite sets.
