You might ask what is the smallest distributive lattice that contains a subgraph homeomorphic (or appropriately morphic) to K5 or K3,3.

As a graph it is planar. You need to specify if you want an orientation of the plane and an order preserving map from the lattice to the plane of representation with additional properties for the induced edges.

Isn't a question like this considered in ray tracing algorithms?

For clarity, I would write "surjective (on V)", so that people don't think of covering all edges.

Nice example. As I understand things, one can switch to having very large or very small ratios, and it can began with any term. I do not see this as a counterexample. I recommend leaving it as a source to inspire others.

It depends on what you want to study, equational theories of ( certain subclasses of ) rings, or certain theories which involve the total or even partial order as it relates to the operations. There are other complexity issues as well, e.g. bounded arithmetic. Samuel Buss is a name associated with this latter study.

Much as I enjoy your explanatory graphics, I find this one misleading. I recommend either plotting T(N) for all N in the range ( and then using a log scale), or instead relabelling and plot integer k against T(2^k). A comparison of this graph with the first modification (T(N) vs log N for N not powers of 2) might serve as a good topic in data representation.

Even though there is a graeco-latin square of order 4, I wonder if there is a connection this problem has with G-L squares (or with the symmetric group S_6 having an unusual automrphism group).

Another snap solution: for 5 people and person A , any 5 arrangements must have at least one person P appearing at least twice on A's left.

One reason to use these forms is that it may be possible to characterize those n for which gcd(n, 2n choose n) > 2. In particular, most composite integers n will not be expected to divide (2n choose n) - 2.

You might note the closed forms of the sums also: that is likely to get more people involved in answering.

There are those who might call a term algebra an absolutely free algebra. I call it a term algebra, the free countably generated algebra in the variety of all structures of a given similarity type.

Relatively free algebra with labelling of the term algebra by congruences. I don't recall seeing a categorification of the process as you have it here, but what you have above is not far removed from constructing the free algebra of a variety from a term algebra.

Thanks for the link. Do you or does Bill have an idea about odd multiperfects? I'm looking for literature which may touch on the idea in mathoverflow.net/questions/134826/… , and anything which has a similar smell to it.

Because of large gaps, the degree can't be one. Based on results for integer coefficient polynomials, my guess is the answer is no.

Not if edge weights are involved.

Let M be the zero matrix except having ones in entries a_ii for i from 1 to r. Any matrix M' with entries within 1/4 of the entries of M should also have rank r or close to it.

Try it with a rank r identity matrix first. I think you will not be able to keep the error bound.

You can use matrices that have at most s rows nonzero. You can probably get close to r/s many summands.