How many 2-edge-colourings of $K_n$ are there? I'm writing a paper on Ramsey Theory and it would be interesting and useful to know the number of essentially different 2-edge-colourings of $K_n$ there are. By that I mean the number of essentially different maps $\chi:E(K_n)\to\{1,2\}$.
Of course, $2^{\binom{n}{2}-1}$ is an (almost trivial) upper bound but, having calculated by hand for a few small values of $n$, this is obviously far too high.
I have found quoted values for $n=16$ and $n=17$ but that's pretty much it. I assume, therefore, that it is known for all smaller values, too, and that would be more than adequate for my purposes. Does anyone know of an explicit formula or "good" bound, or where I could find such a thing?

If you google on something like "clapham easier self-complementary graphs",
you'll be lead to a paper which enumerates self-complementary graphs. The number of isomorphism classes of graphs on $n$ vertices plus the number of isomorphism classes of self-complementary graphs is twice the number you are asking for.
Note that this enumeration was first carried out by R. C. Read; you'll find the citation in Clapham's paper.
