TITLE: Helly theorem + Nerve
QUESTION [11 upvotes]: Consider nerve $\mathcal N$ of a finite set of convex sets in $\mathbb R^n$.
Helly theorem says that $\mathcal N$ is completely determined by its $n$-skeleton, say $\mathcal N_n$.
It seems that not all finite  simplicial complexes with dimension $\le n$ can appear as $\mathcal N_n$.
(For example take 2-dimenisonal simplicial complex which is homeomorphic to $\mathbb{R}\mathrm{P}^2$,
it can not appear as $\mathcal N_3$ for finite set of convex sets in $\mathbb R^3$.)

Is it possible to describe all finite simplicial complexes  which can appear as $\mathcal N_n$?


The question is inspired by a short discussion here.
I am sure a lot should be known, but a quick search gave me nothing.

REPLY [3 votes]: I really wish I knew more about general nerve complexes, so I'm afraid I don't have much to bring to your question. But information on intersection graphs is a lot easier to figure out, so I thought I'd add some remarks to what Gil wrote:
Non-planar Graphs
First, I want to explain how his remark about non-planar graphs works: say you have an edge between two vertices $u$ and $v$ of a graph $G$. Split the edge with a new vertex $e$ to get the new graph $G^\star$. Now, suppose that $G^\star$ is the intersection graph of some arrangement of convex sets in the plane, with sets $U$, $V$ and $E$ that correspond to the vertices of the same name. Then since there are edges $(u,e)$ and $(v,e)$ in $G^\star$, you can pick points $a\in U\cap E$ and $b\in V\cap E$, and the segment $[a,b]$ is of course contained in $E$.
Do this for all vertices of $G^\star$ that arose from adges in $G$. This gives you a collection of edges that must be pairwise disjoint since none of the $e$ vertices are adjacent to each other, they're only adjacent to vertices in $V(G)$. Now, contract the convex sets of type $U$ that correspond to vertices in $V(G)$: this keeps the segments disjoint (except for endpoints) and gives a picture of $G$ on the plane. Thus the convex arrangement fr $G^\star$ exists only if $G$ is planar.
Boxes and Graphs
Also, I think it's important to stress how special interval graphs are. To me, one of the best way to see this is to consider the natural generalization: instead of intervals in $\mathbb{R}$, consider boxes: cartersian products of intervals in $\mathbb{R}^d$. Then, F. S. Roberts proved (in his PhD thesis I think) that any graph can be realized as the intersection graph of a collection of $d$-boxes, and we can even take $d\leq |V(G)|/2$. (The smallest dimension you can choose for a graph $G$ is called the boxicity  of $G$.)
If you'd rather look at more general convex sets, then it's been known for even longer that you can realize a graph as the intersection pattern of convex sets of $\mathbb{R}^3$ (and thus showing some sort of counterpoint to Gil Kalai's answer in dimension 2). The obvious conclusion: nerve complexes carry a lot more information than graphs; nothing new or surprising there, but these examples can help you really appreciate that.
References
Roberts, F. S. (1969), "On the boxicity and cubicity of a graph", in Tutte, W. T., Recent Progress in Combinatorics, Academic Press, pp. 301–310,