{
  "abstract": "Erd\u0151s problem #501 (Erd\u0151s 1961; Erd\u0151s\u2013Hajnal 1971, Problem 38): for every real x let A_x be a bounded set of reals of Lebesgue outer measure < 1; must there be an infinite independent set, i.e. an infinite X \u2286 \u211d with x \u2209 A_y for all distinct x, y \u2208 X?  And if the sets A_x are closed of measure < 1, must there be an independent set of size 3?  This repository formalizes both answers.  Second question: yes \u2014 closed sets of measure < 1 admit an infinite independent set (Newelski\u2013Pawlikowski\u2013Seredy\u0144ski 1987; no boundedness is needed), hence one of size 3.  First question: independent of ZFC.  Under CH (as \u2135\u2081 = \ud835\udd20) Hechler's 1972 construction gives a family of bounded null sets with no infinite independent set \u2014 a Mathlib-level theorem \u2014 and the negation of the first question holds in the Boolean-valued model of the collapse algebra Col(\u03c9\u2081, \ud835\udcab(\u03c9)); after adding \ud835\udd20\u207a random reals (Glazer and Sol 2026, adapting the paper's \u03c9\u2082 random reals over a CH ground) the first question has a positive answer, so it holds in the Boolean-valued model of the random algebra with \ud835\udd20\u207a coordinates.  Both Boolean-valued models are built in a Lean 4 port of Han\u2013van Doorn's Flypitch (the framework of their independence-of-CH proof), and Flypitch's completeness theorem turns them into two-valued models.  The comparator challenge states the results in Mathlib alone: the language of set theory, the theory ZFC (Flypitch's axiomatization: extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, strong collection) and the sentence Erdos501 (\"every complete ordered field has the Erd\u0151s property\") are defined in Mathlib's ModelTheory, and independence is stated semantically as \u00ac (ZFC \u22a8\u1d47 Erdos501) \u2227 \u00ac (ZFC \u22a8\u1d47 \u223cErdos501); a faithfulness theorem shows that in Mathlib's ZFSet the sentence is equivalent to the Mathlib statement of the first question, verbatim the proposition of google-deepmind/formal-conjectures.  All targets are proved from propext, Classical.choice and Quot.sound only.  As far as we know this is the first Erd\u0151s problem whose resolution is a formally verified independence result.",
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        "authors": [
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            "name": "Paul Erd\u0151s"
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        "identifier": "Magyar Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl. 6 (1961), 221\u2013254",
        "location": "Problem II.9",
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        "title": "Some unsolved problems (Problem II.9)",
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            "name": "Paul Erd\u0151s"
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        "authors": [
          {
            "name": "Thomas F. Bloom (ed.)"
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          {
            "name": "Sungchul Lee"
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            "name": "Nat Sothanaphan"
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          {
            "name": "Sol"
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        "authors": [
          {
            "name": "Ludomir Newelski"
          },
          {
            "name": "Janusz Pawlikowski"
          },
          {
            "name": "Wies\u0142aw Seredy\u0144ski"
          }
        ],
        "identifier": "Proc. Amer. Math. Soc. 100 (1987), 335\u2013339",
        "relationship": "formalizes",
        "title": "Infinite free set for small measure set mappings",
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      },
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          {
            "name": "Stephen H. Hechler"
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        "relationship": "formalizes",
        "title": "Directed graphs over topological spaces: some set theoretical aspects",
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        "authors": [
          {
            "name": "Elliot Glazer"
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            "name": "Sol"
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        "identifier": "docs/paper/erdos501_random_profiles_rev10.pdf (unpublished draft, 2026-08-16)",
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        "title": "Erd\u0151s Problem 501 after adding \u03c9\u2082 random reals (draft, revision 10)",
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