There’s lots of great stuff in Ramsey theory. In some sense it feels quite different to a lot of other mathematics. It also has the great property that it is ty

There’s lots of great stuff in Ramsey theory. In some sense it feels quite different to a lot of other mathematics. It also has the great property that it is typically very hard to look at a statement in Ramsey theory and know if it is likely true or not, or even if it will likely be easy to prove or not. For example consider a recently disproven long-standing problem of Ramsey theory: Is it possible to colour the natural numbers with two colours such that no Pythagorean triple is all one colour? This question was asked by Ron Graham many years ago with a $100 prize for the answer, assuming it would be reasonably easy to prove. It was recently solved with an absolutely enormous multi-terabyte computer proof. It turns out you can colour everything up to 7824 fine but no choice for 7825 is sufficient.