Background: I am reading a paper (link) that concerns linear programs (with unknown constraints).
If a linear program has constraints $\mathbf A \mathbf x \leq \mathbf b$, where $\mathbf A \in \mathbb R^{m \times n}$, $\mathbf x \in \mathbb R^n$, and $\mathbf b \in \mathbb R^n$, then this paper defines a polytope $\mathcal P$ to be $\mathcal P := \left\{ \mathbf x \in \mathbb R^n \colon \mathbf A \mathbf x \leq \mathbf b \right\}$.
The paper uses the set of edges of $\mathcal P$, denoted by $E_{\mathcal P}$, and involves an algorithm that depends on the number of edges, written $\left\lvert E_{\mathcal P} \right\rvert$.
Question: What can be said about the relationship between the dimensions related to the polytope, i.e. $m$ and $n$, and the number of edges, $\left\lvert E_{\mathcal P} \right\rvert$?
My attempt: According to this answer, by McMullen's Upper Bound Theorem, if the polytope has $V$ vertices, then we have
\begin{equation} \left\lvert E_{\mathcal P} \right\rvert \leq \binom{V}{2}. \end{equation}
According to this answer, the number of vertices $V$ will be bounded as follows:
\begin{equation} V \leq \binom{m - \lceil n / 2 \rceil}{\lfloor n / 2 \rfloor} + \binom{m - \lfloor n / 2 \rfloor - 1}{\lceil n / 2 \rceil - 1}. \end{equation}
Is this right? What else can be said? If the above is correct, are there books that contain these results? I have been looking at Chapters 15 and 17 of the Handbook of Discrete and Computational Geometry (link), but I don't see these precise results in there.