1090.lean`

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

1 declarations, 1 contain sorry, 0 declared as axioms

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.

Imports (1)

FormalConjecturesUtil

theorem Erdos1090.erdos_1090 Let $k\geq 3$ sorry line 38 source
Source documentation
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.

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Finset, Fin, Real, Subtype, Membership.mem, And, Collinear, GE.ge, Finset.card, AffineSubspace, affineSpan, Eq, Subtype.mk
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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 := by
…
```
Mentions: c