FormalConjectures.ErdosProblems.«774»

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import FormalConjecturesUtil

Erdős Problem 774

Reference: erdosproblems.com/774


open Finset

We call $A\subset \mathbb{N}$ dissociated if $\sum_{n\in X}n\neq \sum_{m\in Y}m$ for all finite $X,Y\subset A$ with $X\neq Y$.

def Set.IsDissociated (A : Set ℕ) : Prop :=
  { S : Finset ℕ | (S : Set ℕ) ⊆ A }.InjOn fun S ↦ ∑ n ∈ S, n

We call $A$ proportionately dissociated if every finite $B\subset A$ contains a dissociated set of size $\gg \lvert B\rvert$.

In other words, there is a (global) $c > 0$ such that every finite $B \subset A$ contains a dissociated set of size $\geq c|B|$.

def Set.IsProportionatelyDissociated (A : Set ℕ) : Prop :=
  ∃ c > (0 : ℝ), ∀ (B : Finset ℕ), (B : Set ℕ) ⊆ A →
    ∃ S ⊆ B, #S ≥ c * #B ∧ (S : Set ℕ).IsDissociated


namespace Erdos774

Is every proportionately dissociated (infinite) set the union of a finite number of dissociated sets?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_774 : answer(sorry) ↔ ∀ (A : Set ℕ), A.Infinite → A.IsProportionatelyDissociated →
    ∃ (T : Set (Set ℕ)), (∀ S ∈ T, S.IsDissociated) ∧ T.Finite ∧ A = ⋃₀ T := by⊢ True ↔
  ∀ (A : Set ℕ), A.Infinite → A.IsProportionatelyDissociated → ∃ T, (∀ S ∈ T, S.IsDissociated) ∧ T.Finite ∧ A = ⋃₀ T
  sorryAll goals completed! 🐙


end Erdos774