Date:
Wed, 03/07/202411:15-13:00
Title: A Ramsey theorem for the reals
Abstract: I will 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 Shelah had shown that a stronger statement is consistent with a forcing construction assuming the existence of large cardinals (in 1985 from an w1-Erdos cardinal, later in 1992 just a strongly Mahlo cardinal). Then in 2018 Raghavan and Todorčević had proved it axiomatically (from a Woodin cardinal, or a strongly compact cardinal, or a precipitous ideal on w1). I will prove it in ZFC.