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!