In sci.physics Gene Ward Smith <genewardsmith@gmail.com> wrote:

The Clarke embedding theorem has as a consequence that the 1+3
pseudo-Riemannian manifolds of general relativity can be embedded in a
2+87 dimensional Minkowski-like space. If we cut the two time
dimensions down to one, so we are looking at 1+87 dimensional space, we
get rid of physically embarassing solutions such as closed time-like
loops. It is "stably causal" in a sense Clarke defines in his paper,
having a global time function increasing on any timelike or null curve.

It seems as if further restrictions on dimensions could lead to rather
subtle modifications of GR. In particular, one might ask what GR looks
like if the manifolds must be embeddable in a 1+25 dimansional space,
for instance. Has this sort of thing been considered?

For a *local* embedding, you need far fewer dimensions. The high numbers
in Clarke's theorem come from demanding global embeddings of complicated
topologies. I don't know of any physical application for such exotic
topologies; at most, a restriction on the embedding dimension might limit
the "sum over topologies" in some approaches to the path integral in quantum
gravity.

(Note also that Clarke's theorem gives upper bounds. I don't think it's
known whether they are "tight" -- the actual bounds might be lower.)