Problem 598

Erdős Problem #598: declared status 'open'. Formalized: yes. Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every $X\subseteq m$ with $\lvert X\rvert=\kappa$ contains subsets of all possible colours? Prize: no. Tags: ramsey theory, set theory.