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import FormalConjecturesUtilErdős Problem 1090
[Er75f] Erdős, Paul,
namespace Erdos1090
Let $k\geq 3$. Does there exist a finite set $A\subset \mathbb{R}^2$ such that, in any $2$-colouring of $A$, there exists a line which contains at least $k$ points from $A$, and all the points of $A$ on the line have the same colour?
Erdős [Er75f] says Graham and Selfridge proved the answer is yes when $k=3$. Hunter has observed that, for sufficiently large $n$, a generic projection of $[k]^n$ into $\mathbb{R}^2$ has this property, by the Hales-Jewett theorem.
@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1090.lean"]
theorem erdos_1090 : answer(True) ↔ ∀ (k : ℕ), ∀ (hk : 3 ≤ k),
∃ (A : Finset (Fin 2 → ℝ)), ∀ (C : A → Fin 2),
∃ (S : Finset (Fin 2 → ℝ)), ∃ (hSA : S ⊆ A),
Collinear ℝ (S : Set (Fin 2 → ℝ)) ∧ S.card ≥ k ∧
(∀ y ∈ A, y ∈ affineSpan ℝ (S : Set (Fin 2 → ℝ)) → y ∈ S) ∧
∃ c, ∀ x (hx : x ∈ S), C ⟨x, hSA hx⟩ = c := ⊢ True ↔
∀ (k : ℕ),
3 ≤ k →
∃ A,
∀ (C : ↥A → Fin 2),
∃ S,
∃ (hSA : S ⊆ A),
Collinear ℝ ↑S ∧
S.card ≥ k ∧ (∀ y ∈ A, y ∈ affineSpan ℝ ↑S → y ∈ S) ∧ ∃ c, ∀ (x : Fin 2 → ℝ) (hx : x ∈ S), C ⟨x, ⋯⟩ = c
All goals completed! 🐙
end Erdos1090