Part III, Ramsey Theory
"Pick your favourite non-silly place to start the induction."
"Hooray, we're done."
"We can prove the theorem first. Then we're going to have a bit of a laugh at the bounds."
"Grzegorczyk is non-trivial to spell." (It's pronounced 'Greg-orr-chick')
"Whereas Van der Waerden's theorem is tall and thin, the Finite Sums theorem is very short and fat."
"If you try to prove it directly it just won't go."
"We're not just using these to have a laugh."
"We showed it's compact: it's a nice place." <
"I'm sorry, I can't draw curly W's - don't worry, that W will never appear again."
"Define a relationship 'twiddle', such that two things twiddle each other if they're close."
"And we should be able to find, by clever maths or something, the following:" Sparse Ramsey Theory (graduate course), Easter 03