Simplicial Pseudomanifolds with Boundary - Bounding number of maximal faces in terms of number of vertices and dimension

Question

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?

Answer 1

To Question 1: there is no upper bound of the form $f(d) \cdot n$, because cyclic polytopes 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. 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.