TITLE: Is Euler characteristic of a simplicial complex upper bounded by a polynomial in the number of its facets ?
QUESTION [16 upvotes]: What is the best upper bound known on the (absolute value of) the 
Euler characteristic of a simplicial complex
in terms of the number of its facets ?
In particular, I am interested in proving or disproving the following:
If $K$ is a simplicial complex with $N$ facets then 
$|\chi(K)| \leq N^{O(1)}.$  
If $K$ is "shellable" then one can show that $|\chi(K)| \leq N.$
As a partial answer, I would be interested in any other subclasses 
of simplicial complexes where the polynomial upper bound holds.

REPLY [5 votes]: If you only care about Cohen-Macaulay complexes (in particular, shellable complexes are Cohen-Macaulay) then the answer is yes.  Let $\Delta$ be a $(d-1)$-dimensional CM complex.  The key is that we should use the $h$-numbers of $\Delta$ instead of its $f$-numbers.  Most importantly: 

The number of facets in $\Delta$ is the sum of its $h$-numbers (for any complex),
$h_d(\Delta) = (-1)^{d-1}\widetilde{\chi}(\Delta)$ (also for any complex), and
$h_j(\Delta) \geq 0$ for all $j$ (for any CM complex).

Thus $$|\widetilde{\chi}(\Delta)| = h_d(\Delta) \leq \sum_{j=0}^dh_j(\Delta) = f_{d-1}(\Delta).$$