2024/05/28 by Tanmay Inamdar, Inamdar, Tanmay
Mathematics · #03E55 #05C55 #05D10 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #Primary 03E02 #Secondary 03E04
paper · pdf · doi:10.48550/arxiv.2405.18431
openalex publication_date 2024/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Sierpiński 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 Todorčević had proved it assuming the existence of large cardinals. We prove it in ZFC. In fact Raghavan and Todorčević proved, assuming more large cardinals, a similar result for a large class of topological spaces. We prove this also, again in ZFC.