A Ramsey theorem for the reals

Abstract

We prove that for every colouring of pairs of reals with finitely-many colours, there is a set homeomorphic to the rationals which takes no more than two colours. This was conjectured by Galvin in 1970, and a colouring of Sierpi{\'n}ski from 1933 witnesses that the number of colours cannot be reduced to one. Previously in 1985 Shelah had shown that a stronger statement is consistent with a forcing construction assuming the existence of large cardinals. Then in 2018 Raghavan and Todor\v{c}evi\'c had proved it assuming the existence of large cardinals. We prove it in $ZFC$. In fact Raghavan and Todor\v{c}evi\'c proved, assuming more large cardinals, a similar result for a large class of topological spaces. We prove this also, again in $ZFC$.