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

Author: raphy October 18, 2022

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 $\mathbb R$ .