598.lean`

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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 598

Reference: erdosproblems.com/598

Imports (1)

FormalConjecturesUtil

def Erdos598.κ Let $\kappa = (2^{\aleph_0})^+$ line 35 source
Source documentation
Let $\kappa = (2^{\aleph_0})^+$. This is the successor cardinal of the continuum.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Cardinal
Proof uses: Order.succ, HPow.hPow, OfNat.ofNat, Cardinal.aleph0
8 instances, coercions and auxiliary constants left out.
```
noncomputable def κ : Cardinal := Order.succ (2 ^ ℵ₀)
…
```
theorem Erdos598.erdos_598 Erdős Problem 598: Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$ sorry line 44 source
Source documentation
Erdős Problem 598: Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every $X \subseteq m$ with $|X| = \kappa$ contains subsets of all possible colours?

depends on sorry what this means
Statement uses: Iff, True, Infinite, Exists, Subtype, Set, Set.Countable, Quotient.out, Erdos598.κ, Eq, Cardinal, Cardinal.mk, Set.image, Set.ofPred, LE.le, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_598 : answer(sorry) ↔
    ∀ (m : Type*) [Infinite m],
    ∃ c : { s : Set m // s.Countable } → κ.out,
    ∀ X : Set m, #X = κ →
    c '' { s : { sub : Set m // sub.Countable } | s.1 ⊆ X } = Set.univ := by
…
```
Mentions: c, Erdos598.κ