A Ramsey theorem for the reals

TL;DR

This paper proves a combinatorial partition theorem for the real numbers, showing that any finite-colour pairing of reals contains a subset homeomorphic to the rationals with at most two colours, resolving a long-standing conjecture in ZFC.

Contribution

The authors establish the conjecture by Galvin from 1970 in ZFC, removing the need for large cardinal assumptions previously used.

Findings

Proved the conjecture in ZFC.

Extended results to a broad class of topological spaces.

Resolved a 50-year-old open problem.

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 $ZFC$.