Perhaps this is what you want: en.wikipedia.org/wiki/…

Is it known whether there is some algorithm telling you whether there is a ring $(G, +, \times)$ for the given abelian group $(G, +)$? My guess is that already for this question, the answer is no; see also mathoverflow.net/questions/92557/…. (This doesn't answer your second question.)

I like "$k$-rainbow" better than "non-($k-1$)-extendible" :) Perhaps a variation on this theme such as "$k$-prismatic" or "$k$-polychromatic" might be a good name?

Of course I don't mind :)

Notice that you are looking for a labeling of the edges by the 15 smallest primes. The Petersen graph has symmetry group $S_5$ so that means that there are, in principle, $15!/5!$ possibilities to check, which is probably still too much to check one by one by computer.

What do you mean? It's literally there: "At first, we denote $|\mathcal{C}|=m$ and do not assume for a moment that $m=n$, but prove that $m \leq n$."

This is shown in the answer to your earlier question mathoverflow.net/questions/266511/….

@ArturoMagidin: This would only be true if you would add the relation $\psi^n=1$ (where $n$ is the order of $\phi$) as indicated by Derek's comment.

I think it's called reflexive rather than reflexible. (Both your own course notes and Peter's notes seem to confirm this.) I've never encountered "reflexible" before, in any case...

The spherical building itself is just a projective space, which is easy to understand. What exactly is it you would like to know?

@MarekMitros: "For me the definition of things is less important." -- What? How can you communicate about mathematics if you don't have precise definitions? The OP points out exactly that having different definitions for the same notion can be very confusing, and should be avoided as much as possible.

Perhaps a better reference than the youtube video is artofproblemsolving.com/wiki/index.php?title=1988_IMO_Proble‌​ms/…. (This is problem 6 from the 1988 IMO.)

Wikipedia says "For larger values of $k$, the number of forbidden minors grows at least as quickly as the exponential of the square root of $k$.[9] However, known upper bounds on the size and number of forbidden minors are much higher than this lower bound.[10]". (en.wikipedia.org/wiki/…) Have you checked this reference?

If $f(x) = g(x)^2$, then both $f(x)$ and its derivative $f'(x)$ are divisible by $g(x)$, so in particular $\gcd(f(x), f'(x))$ is divisible by $g(x)$. Have you tried to compute $\gcd(f(x), f'(x))$? In particular, your assumption that $n=(p^m+1)/2$ is very helpful in that respect, since $2n=1$ in the field $k$.