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 $|X|=\kappa$ contains subsets of all possible colours?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (\mu:=2^{\aleph_0}) and (\kappa:=\mu^+). The question is asking for a coloring [ c:[m]^{\aleph_0}\to \kappa ] such that for every (X\subseteq m) with (|X|=\kappa), [ c\bigl[[X]^{\aleph_0}\bigr]=\kappa, ] i.e. every (\kappa)-sized $X$ is “fully polychromatic” on its countable subsets [[nomath]](in partition notation…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.