  We study Stanley's long-standing conjecture that the h-vectors of matroid
simplicial complexes are pure O-sequences. Our method consists of a new and
more abstract approach, which shifts the focus from working on constructing
suitable artinian level monomial ideals, as often done in the past, to the
study of properties of pure O-sequences. We propose a conjecture on pure
O-sequences and settle it in small socle degrees. This allows us to prove
Stanley's conjecture for all matroids of rank 3. At the end of the paper, using
our method, we discuss a first possible approach to Stanley's conjecture in
full generality. Our technical work on pure O-sequences also uses very recent
results of the third author and collaborators.

Combinatorics;Commutative Algebra