Set-theoretic status of Erdős Problem 501’s first question

For every family (Aₓ) of subsets of R with outer measure below one, must there be an infinite set X such that x∉Aᵧ for all distinct x,y∈X? The question includes whether this assertion can be decided from ZFC.

Why it matters and what counts as progress

Why it matters: Relative independence identifies a set-theoretic limitation on deciding a measure-theoretic free-set assertion from ZFC alone.

What counts as progress: Construct a model satisfying the assertion without large-cardinal assumptions or establish its relative independence alongside the known CH counterexample.

How the problem developed

1960 · Partial result [1]
Paul Erdős and András Hajnal Established arbitrarily large finite independent sets for the first question’s set-mapping assumptions.
1972 · Conditional counterexample [1]
Stephen H. Hechler Showed that the first question has a negative answer assuming the Continuum Hypothesis.
29 May 2026 · Conditional result [2][3]
Sungchul Lee, with GPT-5.5 Pro assistance Reported a positive answer assuming Lebesgue measure extends to a countably additive measure on every subset of the reals.
1 Jun 2026 · Revised proof [2][3]
Sungchul Lee, with GPT-5.5 Pro assistance Revised the note, replacing its appeal to a section inequality attributed to Kunen with a direct elementary proof of the needed inequality.

AI contribution

Erdős Problem 501 after adding ω₂ random reals

Elliot Glazer, Sol

A ZFC model has the property that every family (Aₓ) of subsets of ℝ with outer measure below one admits an infinite X such that x∉Aᵧ for distinct x,y∈X. Combined with Hechler’s earlier CH counterexample, this gives relative ZFC independence for the first question of Erdős 501.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (Lean 4.34.0-rc1 with Mathlib and Flypitch4; Comparator and axiom audits)

Human understandingFew