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Level A · Machine-checkable Hard Combinatorics P-erdos-501

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?

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 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

For every let be a bounded set with outer measure . Must there exist an infinite independent set, that is, some infinite $X \subseteq \mathbb{R}x \notin A_yx \neq y \in X$?

If the sets are closed and have measure , then must there exist an independent set of size ?

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». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_501 : answer(sorry) ↔
    ∀ (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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.