On the $h$-vectors of Cohen-Macaulay Flag Complexes

Starting from an unpublished conjecture of Kalai and from a conjecture of Eisenbud, Green and Harris, we study several problems relating $h$-vectors of Cohen-Macaulay, flag simplicial complexes and face vectors of simplicial complexes.

Face Vectors of Two-Dimensional Buchsbaum Complexes

In this paper, we characterize all possible $h$-vectors of $2$-dimensional Buchsbaum simplicial complexes.

Flag Vectors of Multiplicial Polytopes

Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face vectors. A special class of multiplicial polytopes, also discovered by Bisztriczky, is comprised of the ordinary polytopes. These are a natural generalization of the cyclic polytopes. The flag vectors of ordinary polytopes are determined. This is used to give a surprisingly simple formula for the $h$-vector of the ordinary $d$-polytope with $n+1$ vertices and characteristic $k$: $h_i={k-d+i\choose i}+(n-k){k-d+i-1\choose i-1}$, for $i\le d/2$. In addition, a construction is given for 4-dimensional multiplicial polytopes having two-thirds of their vertices on a single facet, answering a question of Bisztriczky.