The infinite Ramsey theorem (1928) states that for any colouring of pairs of natural numbers (or triples, quadruples etc.) into finitely-many colours, there is an infinite set on which the colouring is constant. Sierpinski proved in 1933 that there is no straightforward analogue of it for the reals. In my talk I will discuss my proof of a conjecture of Galvin from 1970 which yields the optimal Ramsey theorem for the reals.