### Nerves of convex sets in general

There is quite a bit known about nerves of families of convex sets in $R^n$. Indeed Helly's theorem asserts that the $n$-skeleton determines the entire complex. In fact, considerably more is known beyond Helly's theorem. It follows, for example that all homology group of the nerve vanish in dimensions larger or equal to n. Morover, this property is inherited to induced subcomplexes and to links of the nerve. (Because those are also nerves of families of convex sets in the same Euclidian space.)

For a survey see this paper: Martin Tancer, Intersection patterns of convex sets via simplicial complexes, a survey.

However, only little is known about the skeletons of the nerves below dimension n.

Every n dimensional complex can be represented as a nerve of convex sets in $R^{2n+1}$. This was proved by Wegner in 67 and again by Perel'man in 85. (This was Perel'man's first paper.) It is interesting to understand systematically obstructions for nerves of convex sets in $R^n$ whose dimension is between n/2 and n. For n=1 much is known. There are substantial results in the plane but not so many in higher dimensions.

### Dimension 1 - interval graphs

There is a lot known about interval graphs which is what you ask about for n=1. This is a very restricted and well understood class of graphs. It is a subclass of the class of chordal graphs with itself is a very special class of the class of perfect graphs.

### Families of convex sets in the Dimension 2

An example of the kind asked in the question to the best of my memory is obtained as follows: Start with a non planar graph, and divide every edge by adding a vertex in it. Then this graph is not a nerve of conves sets in the plane.

Since the question was about representing n dimensional complexes as nerves of families of convex sets in $R^m$ where $m>n$ let me mention a few specific results in this direction where n=1 and m=2.

Theorem (D. Larman, J. Matousek, J. Pach, J. Torocsik): For a family of planar convex sets either there is a subfamily of $n^{1/5}$ sets which are pairwise intersecting or there is a subfamily of $n^{1/5}$ sets which are pairwise disjoint.

(D. Larman, J. Matousek, J. Pach, J. Torocsik, A Ramsey-type result for convex sets. Bull. London Math. Soc. 26 (1994), no. 2, 132–136.)

A result which also directly follows from this paper is:

For a family of planar convex sets either

there is a family of p sets which are pairwise disjoint or

there is a family of $c_p n$ sets which are pairwise intersecting.

(It is not known if it is possible to replace “pairwise disjoint” with “pairwise intersecting” in this last theorem. Fox and Pach have some results in this direction.)

And the following beautiful theorem:

Theorem (J. Fox, J. Pach and Cs. D. Toth):

Every family of plane convex sets contains two subfamilies of size $cn$ such that:

either each element of the first intersects every element in the second, or

no element in the first intersects any element of the other.

### Rough expected picture

Morally, graphs that are nerves of convex sets in the plane are far in their behavior from random graphs and come close (in a sense) to perfect graphs. (This comment applies to other graphs and hypergraphs arising in geometry.) Such statements (towards perfectness in a weak sense) are known to hold for the $n$-dimensional skeletons of nerves of families of convex sets in $R^n$. We may expect that this phenomena will start occuring (in weakers forms) already for $n/2$-dimensional skeleta but there are very few known results beyond the plane.