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Add Erdős Problem 501 (independent sets for real-indexed families)#3777

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henrykmichalewski:add-problem-501
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Add Erdős Problem 501 (independent sets for real-indexed families)#3777
henrykmichalewski wants to merge 1 commit intogoogle-deepmind:mainfrom
henrykmichalewski:add-problem-501

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@henrykmichalewski henrykmichalewski commented Apr 16, 2026

fixes #758

Problem

Erdős Problem 501: https://www.erdosproblems.com/501

For every family of sets indexed by ℝ, each of positive outer measure, does there exist an uncountable independent set?

Contents

  • IsIndependent definition for families of sets indexed by ℝ
  • Outer measure encoding for positive-measure sets
  • Main open theorem erdos_501
  • 6 variants (Erdős-Hajnal original, Hechler CH result, NPS consistency, Gladysz classical)
  • 2 fully proved lemmas (singleton independent, empty family independent)
  • Sanity checks ensuring definitions are non-trivial

Assisted by Claude (Anthropic).

…families

Adds formalization of Erdős Problem 501 (Erdős-Hajnal).
Reference: https://www.erdosproblems.com/501

Asks whether for every family of sets indexed by ℝ, each of positive outer
measure, there exists an uncountable independent set. Includes IsIndependent
definition, outer measure encoding, 6 variants (Erdős-Hajnal, Hechler CH,
NPS, Gladysz), 2 proved lemmas, and sanity checks. Assisted by Claude
(Anthropic).
@github-actions github-actions bot added the erdos-problems Erdős Problems label Apr 16, 2026
@henrykmichalewski
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Closes #758

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Erdős Problem 501

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