Erdős (Erdos) Problem #501: status in 2026

Erdős Problem #501

For every x ∈ ℝ let A_x ⊂ ℝ be a bounded set with outer measure < 1. Must there exist an infinite independent set, that is, some infinite X ⊆ ℝ such that x ∉ A_y for all x ≠ y ∈ X? If the sets A_x are closed and have measure < 1, then must there exist an independent set of size 3?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-09-03). It is marked as not disprovable from the axioms of ZFC.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #501 as not disprovable (status updated 3 September 2026).
Formal Conjectures marks the Lean statement erdos_501 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #501 on erdosproblems.com	not disprovable 2026-09-03	erdosproblems.com
Formal Conjectures: status of `erdos_501`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 5 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-501,
  title        = {Erdős Problem #501},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-501}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

For every $x \in R$ let $A_{x} \subset R$ be a bounded set with outer measure $< 1$ . Must there exist an infinite independent set, that is, some infinite $X \subseteq \mathbb{R} $s u c h t ha t$ x \notin A_y $f or a l l$ x \neq y \in X$?

If the sets $A_{x}$ are closed and have measure $< 1$ , then must there exist an independent set of size $3$ ?

Known results: Erdős–Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets. Hechler [He72] showed the answer is no assuming the continuum hypothesis.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«501».

theorem erdos_501 :
    ∀ (A : ℝ → Set ℝ),
      (∀ x, Bornology.IsBounded (A x)) →
      (∀ x, volume.toOuterMeasure (A x) < 1) →
      ∃ X : Set ℝ, X.Infinite ∧ X.Pairwise (fun x y => x ∉ A y)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/501. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/501
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221-254.
[ErHa71] Erdős, Paul and Hajnal, András, Unsolved problems in set theory. Axiomatic Set Theory, Proc. Sympos. Pure Math. XIII Part I (1971), 17-48.
[ErHa60] Erdős, Paul and Hajnal, András. On some combinatorial problems involving complete graphs. Acta Math. Acad. Sci. Hungar. (1960), 395-424.
[Gl62] Gladysz, S. Some topological properties of independent sets. Colloq. Math. (1962).
[He72] Hechler, S. H. A dozen small uncountable cardinals. TOPO 72, Lecture Notes in Math. (1972), 207-218.
[NPS87] Newelski, L., Pawlikowski, J., and Seredyński, F. Infinite independent sets in the closed case. Acta Math. Acad. Sci. Hungar. (1987).

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.