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Let $\Delta$ be a simplical complex.

Call $\Delta$ pure if all the maximal faces have the same dimension.

Call $\Delta$ Eulerian if it is pure and $\chi (lk (F))= \chi (S^{dim (lk(F))})$ for any $F \in \Delta$.

Question 1: From what I understand characterizing the $f$-vectors of pure simplicial complexes is hopeless at the moment. What if however I give you an $f$ (or $h$)-vector? What are some necessary conditions that I can check for it to be the $f$-vector of a pure simplicial complex.

Say I give you something like this: $(1,19,99,276, 504, 630, 546, 324, 126, 28)$. One can check the Kruskal-Katona bounds (which hold in this case) and even produce the shifted complex, however this is not generally pure even if the $f$-vector is pure. Are there any similar constructions of these type?

Questions 2: What can one say about the $h$-vectors of Eulerian complexes other that they are symmetric?

Thank you!

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I don't understand your example $f$-vector. Are there $19$ edges and only one vertex? – Vidit Nanda Sep 25 '13 at 1:58
The first entry is usually denoted by $f_{-1}$ and is always $1$ since it counts the empty set in the simplicial complex. So in this case the complex would have $19$ vertices and $99$ edges. – Alexandru Papiu Sep 25 '13 at 4:58

A related question is to obtain information on f-vectors of pure multicomplexes (or order ideals of monomials). Any restriction on them would also be a restriction for simplicial complexes. For information on pure multicomplexes see Chapter 8 of this monograph is on simplicial complexes. There it is explained that a complete characterization is probably hopeless since it would include a characterization of the orders of finite projective planes.

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As Gil remarks in his comment, Corollary 1 of the paper which I mentioned does not in fact imply the upper bound conjecture except when one additionally assumes isolated singularities. Still, I hope that this paper and its references may be of some use to the OP.

At least partial answers to both your questions may be found in the following paper:

Patricia Hersh and Isabella Novik, A short simplicial $h$-vector and the upper bound theorem, Discrete & Computational Geometry, 28 (3): 283-289 (2002).

In particular, the main theorem implies (via Corollary 1) that if the dimension $d$ is odd, then the first $d-1$ entries of your $f$-vector will be bounded above by those of the cyclic polytope $C(n,d)$. For additional details, look for the "upper bound conjecture" as well as the references in this paper.

I don't think much is known when $d$ is even.

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Vidit, the paper by Hersh and Novik deals with Eulerian complexes with isolated singularities, which is a very special case. – Gil Kalai Sep 27 '13 at 3:20
@GilKalai Thank you for noticing that I had overestimated the consequences of that result in Hersh-Novik. I will modify my answer. – Vidit Nanda Sep 27 '13 at 4:07

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