Erdős Problem 1090

References:

• erdosproblems.com/1090
• [Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.

theorem Erdos1090.erdos_1090 : True ↔ ∀ (k : ℕ), 3 ≤ k → ∃ (A : Finset (Fin 2 → ℝ)), ∀ (C : ↥A → Fin 2), ∃ (S : Finset (Fin 2 → ℝ)) (hSA : S ⊆ A), Collinear ℝ ↑S ∧ S.card ≥ k ∧ (∀ y ∈ A, y ∈ affineSpan ℝ ↑S → y ∈ S) ∧ ∃ (c : Fin 2), ∀ (x : Fin 2 → ℝ) (hx : x ∈ S), C ⟨x, ⋯⟩ = c

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.