Try projecting a rectangular simplex onto a regular simplex. Then consider how to transform any simplex into a rectangular simplex.

Using the symmetry d goes to $n_N$/d, you can improve the lower bound by dividing the exponent by 2, as well as getting an upper bound by averaging over the interval $(k, N_n^{1/2})$. With examples like 715/714, it may be that you can't remove log k from the exponent.

hmm. Two unit tetrahedra seem to cover a cube of side length root(2)/3. Add one for each face and edge of the unit cube, and I get 34 total.

If you take two tetrahedra, put them base to base, twist one 60 degrees, then smash them together, you might get something that covers a cube of side length close to 1/2. In any case, I'm guessing the required number of tetrahedra is close to 20.

True, but the latter question has a nice candidate based on the height h of the n+1 simplex. If c_n is the side length for a cube in scribed in an n simplex, a lower bound for c_n+1 (=c) comes from the relations c +rh=h and c= rc_n.

You might try an upper bound based on inscribing a cube inside a unit simplex.

Good point. I know of literature that talks about a/b words for some value of a/b at least 5/2. Any literature that might address your problem should have Zimin in the bibliography if not in the text.

Check out Zimin words.

For contrast and guidance, check out mathoverflow.net/q/30661

I count diameter as length of a path from end-to-end, counted by vertices. But I am not a graph theorist. Also, this just shows small diameter for a small initial section of the graph. However, one more step from b may reach the rest of the graph.

Does "mathematical" rule out "search for unified field theory"?

@Emil, I don't see anything wrong either, and I suspect things will turn out OK for the OP. But then, I am not in academia, and have little or no clue what the ramifications are to someone who may be looking for a postdoc or other academic position. Ideally, there should be no ramifications, but I sense that Pace is touching upon such in his answer.

@Wolfgang <a href='mathoverflow.net/q/201374/6094' >Good Question!</a>

@Joel : Andreas, or Emil? (Did Andreas delete a post near this question?)

The arxiv is an indexed repository with a very small amount of quality control, not a journal. Placing something in arxiv makes it more accessible than putting it on some other website because many look to arxiv to see what it has. However, it is a place to expose ones flaws as well as ones strengths. I think the real question is not whether to place it, but what are the downsides to making something available that is not journal quality. In sharing information, should the phrase "journal quality" even matter? Or the phrase "public perception"? Or even "personal/career reputation"?