How is the minimum embedding dimension of an abstract simplicial complex related to its simplices? I was working on an example of an abstract simplicial complex and its geometric realization and I had a question. I think the question makes more sense when explained through the example. Some definitions before the example:
An abstract n-simplex, $\Delta_n$, is the set of all non-empty subsets of {0,...,1}.
The geometric standard n-simplex, $|\Delta_n|$ = {$x =(x_0,...,x_n) \in \mathbb{R}^{n+1} | \sum_{i=0}^{n} x_i = 1$}.
A geometric realization of $\Delta_n$ is $|\Delta_n|$ as a subspace of $\mathbb{R}^{n+1}$.
Now for the example, let $V$ be the vertex set {$A,B,C,D$} with 1-simplices {$AB,AC,AD,BC$} and 2-simplices {$ABC$}.
This simplex can be embedded in $\mathbb{R}^2$. Here I mean embedded in the sense that it can be realized as a set of points in $\mathbb{R}^2$. But when {$BCD$} is added, the simplex can no longer be represented in $\mathbb{R}^2$. See here,

My question is, given a $\Delta_n$ that can be embedded in $\mathbb{R}^m$ (where $m$ is the minimum dimension possible), what simplices can be added while maintaining the property that $\Delta_n$ can be embedded in $\mathbb{R}^m$. Does it have to do with the size of the simplex being added or does it deal with the number of simplices being added? Is there even any connection between the minimum possible embedding dimension and the type/number of simplices?
Side note: I brought this up to my professor and he said it might be related to when graphs are planar.
 A: Thanks for your curiosity and nice pictures.
I don't have much time but I'm working on similar stuff right now so I'll drop something.
I think your notation is a bit off.
Your $V$ is not a simplex. Note that $\Delta_n$ is intented to be a fixed object that only depends on $n$.

Does it have to do with the size of the simplex being added or does it deal with the number of simplices being added? Is there even any connection between the minimum possible embedding dimension and the type/number of simplices?

It has to do as well with the size (I assume by this you mean dimension) of the simplices you add, the number and their configuration.
A simplicial complex with $n$ vertices can always be embedded in $\mathbb{R}^n$.
Menger proved in 1928 that every $d$-dimensional simplicial complex can be embedded in $\mathbb{R}^{2d+1}$.
Indeed, embedding twodimensional complexes in $\mathbb{R}^3$ is related to planar graphs, because in general we have the following.
Let $\Sigma$ be a simplicial complex and $\mathbf{v}$ a vertex of $\Sigma$.
Consider the subcomplex $\Sigma'$ of $\Sigma$ consisting of all those cells of $\Sigma$ that donot contain $\mathbf{v}$.
If $\Sigma'$ can be embedded in $\mathbb{R}^d$, then $\Sigma$ can be embedded in $\mathbb{R}^{d+1}$.
Substantial work has been done on understanding which complexes can be embedded in which dimensions of Euclidean space. But it's not easy.
You might want to have a look at the following freely available papers:

*

*On embedding polyhedra and manifolds by Kreso Horvatic
https://www.ams.org/journals/tran/1971-157-00/S0002-9947-1971-0278314-4/S0002-9947-1971-0278314-4.pdf

*Embedding 2-complexes in $\mathbb{R}^4$ by Marko Kranjc https://www.semanticscholar.org/paper/EMBEDDING-2-COMPLEXES-IN-R4-Kranjc/2bdd0fb94b8a8891d8a56fc17476f12e9c5bd916
