Here are two amazing physical simulations (using ball bearings!) of the Lubachevsky-Graham-Stillinger model, by Finnish artists Tommi Grönlund and Petteri Nisunen. vimeo.com/37782969 dump.com/giantsurface

Dear Timothy, I have a small bone to pick with this answer. While graph minor theory is indeed a grand project, the proof of the strong perfect graph conjecture and the characterization of the structure of claw-free graphs are not part of graph minor theory. Both are concerned with forbidden induced subgraphs, rather than forbidden minors. The tools used in studying forbidden induced subgraphs are rather different, as witnessed by the fact that the paper containing the proof of the strong perfect graph conjecture does not reference a single paper from the graph minors sequence.

(First paragraph, I mean.)

Aldous points out in the first line of this article (stat-www.berkeley.edu/~aldous/Papers/me49.pdf) that the distribution is not the uniform distribution.

I checked the Moon Moser paper, that is indeed your reference. I've emailed you a somewhat crappy but legible scan of the paper.

There are results of Alon (tinyurl.com/nogapaper) and of Bollobas and Sarkar (myweb.facstaff.wwu.edu/sarkara/four.ps) on maximizing the number of copies of P_4 over graphs with a fixed number of edges. Not posting as an answer since the word "induced", and fixing the number of edges rather than of vertices, makes a pretty big difference. As a historical curiosity, this seems to be Noga Alon's first paper, according to the publication list on his web site.