The two versions of the problem are fully equivalent. Suppose $X$ is a finite set and $\mathbf{P}$ is a collection of subsets of $X$. Let's define
$$f(\mathbf{P})\mathrel{\mathop:}=\sum_{S\in \mathbf{P}}(-1)^{|S|} \qquad \text{and} \qquad g(\mathbf{P})\mathrel{\mathop:}= \sum_{S_1,S_2,\dots,S_r\in \mathbf{P}\\ S_1\cup \cdots \cup S_r=X}(-1)^r.$$
Let's also denote by $\widehat {\mathbf{P}}$ the set of all subsets which contain some element of $\mathbf{P}$. The following holds:
$$g(\mathbf{P})=g(\mathbf{\widehat{\mathbf{P}}})=f(\widehat{\mathbf{P}}).$$
To prove the first equality notice that if $A_0\subset A_1$ are subsets such that $A_0\in \mathbf{P}$ and $A_1\notin \mathbf{P}$ then
$$g(\mathbf{P}\cup\{A_1\})-g(\mathbf{P})=\sum_{S_1,S_2,\dots,S_r\in \mathbf{P}\\ A_1\cup S_1\cup \cdots \cup S_r=X}(-1)^{r+1}$$
however the collections that index the sum on the right split into those that contain $A_0$ and those that don't. These two cancel each other out and the sum evaluates to zero. Since we can keep adding subsets to $\mathbf{P}$ one by one, this shows that $g(\mathbf{P})=g(\widehat{\mathbf{P}})$. Finally, the equality $g(\widehat{\mathbf{P}})=f(\widehat{\mathbf{P}})$ was proven by Fedor in the previous question (sidenote: this is referred to as Rota's crosscut theorem).

A third equivalent formulation is to ask for bounds on the Euler characteristic of the simplicial complex obtained by using $X$ as a set of vertices and adding a simplex for $S$ whenever the complement of $S$ is in $\widehat{\mathbf{P}}$. Thus your questions become:

- What is the largest Euler characteristic of a simplicial complex with $N$ facets?
- What is the largest Euler characteristic of a simplicial complex with $N$ facets and $m$ vertices?
- What is the largest Euler characteristic of a simplicial complex on $m$ vertices if all facets have dimension $\geq m-l$

The answer to Q1 is $\binom{N-1}{ \lfloor (N-1)/2 \rfloor}$ by the Sagan-Yeh-Ziegler paper. They construct a simplicial complex with $N$ vertices, $\binom{N}{\lfloor N/2\rfloor}$ facets, with Euler characteristic \binom{N-1}{ \lfloor (N-1)/2 \rfloor}, which also gives a simplicial complex with the same Euler characteristic but $N$ facets and $\binom{N}{\lfloor N/2\rfloor}$ vertices. The answer to Q2 was conjectured to be $e^{O(\log N\log m)}$ by David Speyer here, and I don't know what the status of this is.

For Q3, if $m-l\le \frac{m}{2}$ then we can use the same example in Q1 which gives the answer $\binom{m-1}{\lfloor (m-1)/2\rfloor}$. If $m-l\geq m/2$ then the number of facets is at most $\binom{m}{l}$ and assuming Speyer's conjecture the correct upper bound should be $e^{O(\log m \cdot \log \binom{m}{l})}$.

allsubsets of $X$ of cardinality $\leq l$. $\endgroup$ – H A Helfgott Jul 3 at 14:47