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Erdős ProblemsCombinatorics

Erdős Problem 701

Let F\mathcal{F} be a family of sets closed under taking subsets (i.e. if B⊆A∈FB\subseteq A\in\mathcal{F} then B∈FB\in \mathcal{F}). There exists some element xx such that whenever F′⊆F\mathcal{F}'\subseteq \mathcal{F} is an intersecting subfamily we have $$\lv...

CombinatoricsAMS 5

Mathematical statement

Let F\mathcal{F} be a family of sets closed under taking subsets (i.e. if B⊆A∈FB\subseteq A\in\mathcal{F} then B∈FB\in \mathcal{F}). There exists some element xx such that whenever F′⊆F\mathcal{F}'\subseteq \mathcal{F} is an intersecting subfamily we have ∣F′∣≤∣{A∈F:x∈A}∣.\lvert \mathcal{F}'\rvert \leq \lvert \{ A\in \mathcal{F} : x\in A\}\rvert.

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_701

Canonical source
Complete statement target, proof intentionally absentLean 4
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 }) := by  sorry
Statement source
Formal Conjectures
Lean version
v4.27.0
Placeholder
Present; no proof artifact
Source evidence
Pinned source index
Fidelity review
Community formulation

References

Pinned Formal Conjectures sourceformalizationerdosproblems.com/701background