In 1973, Richard Thompson considered the question of whether his newly defined group $F$ was amenable. The motivation for this problem stemed from his observation --- later rediscovered by Brin and Squire --- that $F$ did not contain a free group on two generators, thus making it a candidate for a counterexample to the von Neumann-Day problem. While the von Neumann-Day problem was solved by Ol'shanskii in the class of finitely generated groups and Ol'shanskii and Sapir in the class of finitely presented groups, the question of $F$'s amenability was sufficiently basic so as to become of interest in its own right. In this talk, I will analyze this problem from a Ramsey-theoretic perspective. In particular, the problem is related to generalizations of Ellis's Lemma and Hindman's Theorem to the setting of nonassociative binary systems. The amenability of $F$ is itself equivalent to the existence of certain finite Ramsey numbers. I will also discuss the growth rate of the F\olner function for $F$ (if it exists). Tea at 3:30 pm in SC 1425.