Erdős Problem #501: infinite independent sets for families of small outer measure

Independent of ZFC, which is why this entry is the first to carry that result rather than proved or disproved. Both directions are formalized: Hechler's 1972 construction gives a model where the answer is no, and adding $\mathfrak{c}^+$ random reals over a model of CH gives one where it is yes. The credit is shared and mostly human. Newelski, Pawlikowski and Seredynski settled the problem's second question in 1987, and it is formalized here without the boundedness hypothesis. Hechler supplied one direction in 1972. Sungchul Lee derived a positive answer from a real-valued measurable cardinal, assisted by GPT-5.5 Pro, and Nat Sothanaphan observed that the two halves together give independence. What Glazer and Sol added is the removal of the large cardinal. erdosproblems.com still lists #501 as open at the time of writing.

Exact FrontierDelta

Prior state unknown→independent

Scope and record

Occurred: Aug 19, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-verified. Publication: announcement. AI contribution: ai-co-developed. Imported under CC BY 4.0.

Canonical aliases: Erdős Problem #501: infinite independent sets for families of small outer measure · Erdős #501 · Problem 501

Confidence: Not scored

Registry verification: lean verified · announcement · resolved

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Attribution

VibeMathed
registry · event recorded by

Elliot Glazer
human · human collaborator

Sol
model · ai model contributor · OpenAI

Claude
model · ai model contributor · Anthropic

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Erdős Problem #501: infinite independent sets for families of small outer measure

specialist registry · passed

Both halves of the top rung are present, and both were checked here rather than taken from the announcement. Kernel side: CI was green at the audited commit, Solution.lean contains no sorry and declares no axioms, and the only sorries in the repository are the deliberate placeholders in the two Challenge statement files, which is how a comparator challenge is meant to look. The repo ships an axiom audit recording every one of the seven targets as depending on propext, Classical.choice and Quot.sound alone, with no native_decide and no project axioms. Statement side: the development proves erdos501_sentence_faithful, that its sentence is equivalent in Mathlib's ZFSet to the Mathlib statement of the first question. That claim was checked against the source rather than believed: the target uses Bornology.IsBounded (A x), volume.toOuterMeasure (A x) < 1 and X.Pairwise (fun x y => x ∉ A y), and FormalConjectures/ErdosProblems/501.lean states erdos_501 with the same three predicates in the same shape. Formal Conjectures is a community-reviewed repository, so the statement is anchored outside the proof's own authors. The independence itself is stated semantically, as the existence of models of ZFC on both sides, and restated a second time in proof-theoretic terms via a Lean port of Flypitch.

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Erdős Problem #501: infinite independent sets for families of small outer measure

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Palomar registry listing

witness · pending

palomar link supplied by the VibeMathed registry.

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Formal Conjectures: the upstream statement the faithfulness target is checked against

formal registration · pending

lean-statement link supplied by the VibeMathed registry.