How not to prove Fermat's last theorem

In 1916, Frank Ramsey was 13 years old, and Issai Schur was thinking about Fermat's last theorem. In doing so, he accidentally proved an important special case of Ramsey's theorem, and helped establish at least two major branches of modern combinatorics. In this talk, I'll explain what Schur was up to, and how a combinatorial result like Ramsey's theorem implies an important number-theoretic fact, that Fermat's last theorem cannot be proven using a local-to-global approach. Along the way, we'll encounter one of my favorite open problems, as well as likely the worst math paper I've ever seen.