Colloquium: Tuesday, October 25, 2022. Speaker: Menachem Kojman (Ben-Gurion). Title: “Positive Ramsey relations on uncountable sets”.

Ramsey’s theorem, \mathbb N\rightarrow (\mathbb N)^2, holds for the countable infinity and has many useful applications, but, as Sierpinski discovered, fails for the real line: \mathbb R\nrightarrow (\mathbb R)^2. Strong anti-Ramsey relations are known on many uncountable cardinals and have application in general topology.  

In the talk we shall survey some anti-Ramsey relations on the real line and on other uncountable cardinals and then describe a scheme for weakening  Ramsey’s positive relation by relativizing it to a partition of pairs.  Via this scheme, new positive Ramsey relations are shown  to hold consistently  at various  uncountable cardinals. Some of their instances related to the metric topology of \mathbb R.