the hales--jewett theorem is one of the pillars of ramsey theory, from which many other results follow. a celebrated result of shelah from 1988 gives a significantly improved bound for this theorem. a key tool used in his proof, now known as the cube lemma, has become famous in its own right. hoping to further improve shelah's result, more than twenty years ago, graham, rothschild and spencer asked whether there exists a polynoimal bound for this lemma. in this talk, we present the answer to their question and discuss numerous connections of the cube lemma with other problems in ramsey theory. joint work with david conlon (oxford), jacob fox (mit), and benny sudakov (eth zurich)