Greg, what is your evidence for n=5? (also n=7 and 9 and 15 if you have it.)

The set S is fixed. For each z we construct the sum g which depends on z and S. If n were prime we could use some symmetry, but g has to be bounded even for n not prime and arbitrary z.

I'm still getting my head around the definition. Anyway, take m = 1/epsilon many disjoint cycle graphs of diameter > 2 , and pick a point on each cycle and identify all those m points to get an m petalled flower graph. Does this help with the question? (I'm unsure if it is minimal.)

Since there are fewer than $2^{1250}$ graphs on 50 vertices, I assume we are talking at cross purposes. Either that, or you are counting something different with that $10^{-44000}$.

On m = R(5,5)-1 nodes, what percentage of graphs avoid a 5-clique and an indepdent set of 5 vertices? Is it like over half or fewer than half of all isomorphism types on m nodes? Or is it a very small percentage, as is the case with R(4,4) (I think)? (I guess it should be small, as random search would have pushed the lower bound up by now.)

What kind of pictures does it have?

It looks like latin squares and a generalization of such arrays when d has k distinct prime factors and one has k tuples.