Let $X$ be a set of cardinality ${\aleph }_{\omega }$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\notin A$ for all $A$. Must there exist an infinite $Y\subseteq X$ that is independent - that is, for all finite $B\subset Y$ we have $f(B)\notin Y$?

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

In fact, your question is exactly **Erdős Problem #623**: given (|X|=\aleph_\omega) and (f:[X]^{<\omega}\to X) with (f(A)\notin A) for every finite $A$, must there be an infinite (Y\subseteq X) with (f(B)\notin Y) for all finite (B\subseteq Y)? The current status is listed as **OPEN**. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

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