  The set of f-vectors of pure simplicial complexes is an important but little
understood object in combinatorics and combinatorial commutative algebra.
Unfortunately, its explicit characterization appears to be a virtually
intractable problem, and its structure very irregular and complicated. The
purpose of this note, where we combine a few different algebraic and
combinatorial techniques, is to lend some further evidence to this fact.
  We first show that pure (in fact, Cohen-Macaulay) f-vectors can be
nonunimodal with arbitrarily many peaks, thus improving the corresponding
results known for level Hilbert functions and pure O-sequences. We provide both
an algebraic and a combinatorial argument for this result. Then, answering
negatively a question of the second author and collaborators posed in the
recent AMS Memoir on pure O-sequences, we show that the Interval Property fails
for the set of pure f-vectors, even in dimension 2.
