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Level A · Machine-checkable Hard Combinatorics P-erdos-701

Erdős Problem #701

Let F be a family of sets closed under taking subsets (i.e. if B⊆ AinF then B∈ F). There exists some element x such that whenever F'⊆ F is an intersecting subfamily we have lvert F'rvert ≤ lvert A∈ F : x∈ Arvert.

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-701,
  title        = {Erdős Problem #701},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-701}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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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

Let be a family of sets closed under taking subsets (i.e. if then ). There exists some element such that whenever is an intersecting subfamily we have

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«701». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_701 : answer(sorry) ↔ ∀ {X : Type} [Nonempty X] [Fintype X],
    ∀ (F : Set (Set X)), IsLowerSet F →
      ∃ x : X, ∀ᵉ (F' ⊆ F),
        F'.Intersecting →
          (#F' ≤ #{ A : Set X | A ∈ F ∧ x ∈ A })

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/701. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/701

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.