# 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 $\mathbb R$.

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