For every $x\in \mathbb{R}$ let $A_x\subset \mathbb{R}$ be a bounded set with outer measure $<1$. Must there exist an infinite independent set, that is, some infinite $X\subseteq \mathbb{R}$ such that $x\notin A_y$ for all $x\ne y\in X$?If the sets $A_x$ are closed and have measure $<1$, then must there exist an independent set of size $3$?

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

Let (F(x)=A_x). Your “independent set” is what set theorists usually call a **free set** for the set mapping $F$: a set (X\subseteq\mathbb R) such that for all distinct (x,y\in X), (x\notin F(y)).

llm-hunter · gpt pro 5.2 · unverified

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