Erdős Problem #501 — For every $x\in\mathbb{R}$ let $A_x\subset \mathbb{R}$ be a bounded set with outer measure $<1$.

The problem asks whether every family of bounded sets $A_x\subseteq\mathbb{R}$ with outer measure below $1$ contains an infinite set $X$ whose distinct members never lie in one another’s assigned sets. The unrestricted question remains sensitive to set-theoretic assumptions.

Known results

• Erdős and Hajnal proved the existence of arbitrarily large finite independent sets.
• Hechler (1972) showed that, assuming the continuum hypothesis, the answer to the unrestricted infinite question is negative.
• Newelski, Pawlikowski, and Seredyński (1987) proved a positive result when every $A_x$ is closed.

Conditional proof proposed

A recent forum contribution claims that, assuming Lebesgue measure extends to a countably additive measure on all subsets of the real line, every family with $m^*(A_y)<1$ has an infinite independent set. The argument invokes Fremlin’s Theorem 543C and a Fubini-type inequality attributed to Kunen; it was developed with assistance from GPT-5.5 Pro and remains explicitly unchecked by experts.

Current status (as of March 2026): The closed-set case and the continuum-hypothesis negative result are settled; the original unrestricted problem has only an unverified conditional positive proof and remains open unconditionally.