FormalConjectures/ErdosProblems/965.lean

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L4DEL
Lean4 Domain Explanation Language

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 965

For every 2-coloring of ℝ, is there an uncountable set $A ⊆ ℝ$ such that all sums $a + b$ for $a, b ∈ A, a ≠ b$ have the same colour?

References:

Imports (1)
Imported by 1 file
  1. theorem Erdos965.erdos_965 sorry line 47 source

    depends on sorry what this means

    Statement uses: Iff, False, Real, Fin, OfNat.ofNat, Nat, Exists, Set, And, Not, Set.Countable, Membership.mem, Ne, Eq, HAdd.hAdd

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    8 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

  2. theorem Erdos965.erdos_965.variants.generalization For every 2-coloring of ℝ, is there some $k ≥ 2$ and an uncountable set $A ⊆ ℝ$ such that all sums $a_1 + \cdots + a_k$ of $k$ distinct elements of $A$ have the same colour? sorry line 61 source

    Source documentation

    For every 2-coloring of ℝ, is there some $k ≥ 2$ and an uncountable set $A ⊆ ℝ$ such that all sums $a_1 + \cdots + a_k$ of $k$ distinct elements of $A$ have the same colour?

    In fact, in both [Ko16] and [SWCol] a single 2-coloring of ℝ is constructed such that for every $k ≥ 2$ and every uncountable $A ⊆ ℝ$ the sums of $k$ distinct elements of $A$ are not monochromatic.

    depends on sorry what this means

    Statement uses: Iff, False, Real, Fin, OfNat.ofNat, Nat, Exists, And, GE.ge, Set, Not, Set.Countable, Finset, LE.le, Eq, Finset.card, Finset.sum, id

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    12 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_965.variants.generalization : answer(False) ↔
    ∀ f : ℝ → Fin 2, ∃ k ≥ 2, ∃ A : Set ℝ, ¬ A.Countable ∧ ∀ s t : Finset ℝ,
      ↑s ⊆ A → ↑t ⊆ A → s.card = k → t.card = k → f (s.sum id) = f (t.sum id) := by
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