A simplicial pseudo-manifold of dimension $d$ with boundary is a simplicial complex satisfying the following conditions. 


1. Every maximal face has dimension $d$
2. Each face of dimension $d-1$ is a face of at most two maximal faces.
3. For each two distinct maximal faces $A$ and $A'$, there is a sequence $A\_1,...,A\_r$ of maximal faces such that $A\_1=A$, $A\_r=A'$, and for each $i$, $A\_i$ and $A\_{i+1}$ have a common face of dimension $d-1$.


**Question 1:** Is there a nontrivial upper bound on the number of maximal faces in a simplicial pseudomanifold with boundary in terms of number of vertices and dimension? In particular, is the number of maximal faces in a pseudomanifold with boundary bounded by $f(d)\cdot n$, where n is the number of vertices, and $f(d)$ some function depending only on the dimension? 


Trivially, any $d$-dimensional simplicial complex has at most 
$\binom{n}{d+1} = n^{O(d)}$ faces of dimension $d$. What I would like to know is whether this can be improved to $f(d)\cdot n$ in the special case
of $d$-dimensional pseudomanifolds with boundary.


**Question 2:** Is there an upper bound for the euler characteristic of a pseudomanifold with boundary in terms of dimension? What about pseudomanifolds without boundary? 



Simplicial Pseudomanifolds with Boundary - Bounding number of maximal faces in terms of number of vertices and dimension
To Question 1: there is no upper bound of the form $f(d) \cdot n$, because [cyclic polytopes](https://en.wikipedia.org/wiki/Cyclic_polytope) have $O(n^{\lfloor\frac{d}{2}\rfloor})$ facets.


Cyclic polytopes maximize the number of faces of any dimension among triangulated spheres with $n$ vertices. This is called the [Upper Bound Theorem](https://en.wikipedia.org/wiki/Upper_bound_theorem "Upper bound theorem"). There are works on the UBT for certain classes of pseudomanifolds.


To Question 2: there cannot be an upper bound in terms of dimension. Take a PL ball with many vertices on the boundary. Its Euler characteristic is $1$. Now choose a pair of edges on the boundary and identify them. This increases the Euler characteristic by $1$. This can be repeated many times. The same can be done with a sphere instead of a ball, which yields a pseudo-manifolds without boundary.


