603.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 603

References:

erdosproblems.com/603

Imports (1)

FormalConjecturesUtil

theorem Erdos603.erdos_603 Let $(A_i)$ be a family of countably infinite sets such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$ sorry line 43 source
Source documentation
Let $(A_i)$ be a family of countably infinite sets such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$. Find the smallest cardinal $C$ such that $\cup A_i$ can always be coloured with at most $C$ colours so that no $A_i$ is monochromatic.
GPT 5.4 Pro (prompted by Chojecki) proved there is no uniform bound on such $C$: for every cardinal $C$ there is a family of countably infinite sets $(A_i)$ such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$, and in any colouring of $\cup A_i$ with $C$ colours some $A_i$ is monochromatic.

depends on sorry what this means
Statement uses: Cardinal, Exists, Set, And, Membership.mem, Eq, Cardinal.mk, Cardinal.aleph0, Set.Pairwise, Ne, Inter.inter, OfNat.ofNat, Nat, Set.sUnion, Set.univ, Quotient.out
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_603 :
    ∀ C : Cardinal.{0}, ∃ (V : Type) (F : Set (Set V)),
    (∀ A ∈ F, Cardinal.mk A = ℵ₀) ∧
    F.Pairwise (fun (A B : Set V) ↦ Cardinal.mk ↥(A ∩ B) ≠ 2) ∧
    ⋃₀ F = Set.univ ∧
    ∀ c : V → C.out, ∃ A ∈ F, ∀ x ∈ A, ∀ y ∈ A, c x = c y := by
…
```
Mentions: c