Set mappings on $\mathbb{R}$ with outer measure $<1$: must an infinite free set exist? (Erdős #501)
Statement
For every $x\in\mathbb{R}$ let $A_x\subset \mathbb{R}$ be a bounded set with outer (Lebesgue) measure $<1$. Must there exist an infinite independent set — some infinite $X\subseteq \mathbb{R}$ such that $x\not\in A_y$ for all $x\neq y\in X$? (Secondary question, since resolved — see background: if the sets $A_x$ are closed with measure $<1$, must there exist an independent set of size $3$?) The live open problem is the first question in ZFC: Hechler showed the answer is no assuming the continuum hypothesis, so the question is whether a positive answer is consistent with ZFC, or whether ZFC outright refutes it.
Acceptance. FULLY RESOLVES: settle the status of the first question in ZFC by either (a) constructing in ZFC alone a family $(A_x)_{x\in\mathbb{R}}$ of bounded sets of outer measure $<1$ with no infinite independent set (making Hechler's CH hypothesis unnecessary), or (b) proving the consistency with ZFC of 'every such family admits an infinite independent set' via an explicit forcing model or known consistent axiom (e.g. MA$+\neg$CH), with the metatheoretic hypotheses clearly stated. Machine-checkable (Lean/Coq) proof preferred — a Lean statement already exists in the formal-conjectures repository — else a complete written proof. ADVANCES: prove the answer is decided by a named cardinal characteristic of the continuum (e.g. non(N) or cov(N)), reducing the problem to known independence results; extend the Newelski–Pawlikowski–Seredyński theorem from closed sets to a wider definable class ($F_\sigma$, Borel, analytic) of sets of measure $<1$; or show that under some axiom weaker than CH Hechler's negative construction still goes through. Deliver the proof file (or Lean sources) with all set-theoretic assumptions explicit.
Background
A problem of Erdős from [Er61], revisited by Erdős and Hajnal [ErHa71]; listed as open on erdosproblems.com/501 (fetched 2026-07-13, status 'open'). It belongs to the Erdős–Hajnal theory of set mappings and free sets, here in a measure-theoretic setting. The frontier: Erdős and Hajnal [ErHa60] proved that under the hypotheses of the first question there are arbitrarily large FINITE independent sets, so the whole content is the jump to an infinite one. Hechler [He72] showed that, assuming the continuum hypothesis, the answer to the first question is NO — there is a CH-construction of bounded sets of outer measure $<1$ admitting no infinite independent set. Hence the first question cannot have a positive ZFC answer; what remains open is whether a positive answer is consistent (e.g. under MA$+\neg$CH or in a suitable forcing extension), or whether a negative example exists in ZFC alone. The second question is essentially closed: Gladysz [Gl62] proved independent sets of size $2$ exist, and Newelski, Pawlikowski, and Seredyński [NPS87] proved that if all $A_x$ are closed with measure $<1$ then an INFINITE independent set exists, a strong affirmative answer — the closed case is thus fully understood, and the residual problem is the gap between closed sets and general bounded sets of outer measure $<1$. The statement has been formalised in Lean in Google DeepMind's formal-conjectures repository. The attacker's tool: this is a set-theory problem — forcing constructions (random/Cohen models), cardinal-characteristic analysis (which invariants of the continuum decide the answer), and transfinite constructions under axioms beyond CH; there is no computational purchase.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #501 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #501 (formal-conjectures, Google DeepMind) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.