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Level A · Machine-checkable Hard Graph theory P-erdos-75

Erdős Problem #75

Is there a graph of chromatic number ℵ_ 1 with ℵ_ 1 vertices such that for all ε > 0, if n is sufficiently large and H is a subgraph on n vertices, then H contains an independent set of size > n ^ (1 - ε)?

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-75,
  title        = {Erdős Problem #75},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-75}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims
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Verified
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Disputed
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Refuted
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On the literature board
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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

Is there a graph of chromatic number ℵ_ 1 with ℵ_ 1 vertices such that for all ε > 0, if n is sufficiently large and H is a subgraph on n vertices, then H contains an independent set of size > n ^ (1 - ε)?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«75». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_75 :
    answer(sorry) ↔
    ∃ (V : Type) (G : SimpleGraph V),
      G.chromaticCardinal = ℵ_ 1 ∧
      #V = ℵ_ 1 ∧
      ∀ ε > (0 : ℝ),
        ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (H : G.Subgraph),
            H.verts.ncard = n →
            ∃ (I : Finset V),
              (I : Set V) ⊆ H.verts ∧
              G.IsIndepSet (I : Set V) ∧
              (I.card : ℝ) > (n : ℝ) ^ (1 - ε)

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/75. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • [erdosproblems.com/75] (https://www.erdosproblems.com/75)

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.