214.lean`

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

10 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 214

References:

erdosproblems.com/214
[CsTo94] Csizmadia, György and Tóth, Géza, Note on a Ramsey-type problem in geometry. J. Combin. Theory Ser. A (1994), 302-306.
[EGMRSS75] Erdős, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean Ramsey theorems. II. (1975), 529-557.
[Ju79] Juhász, Rozália, Ramsey type theorems in the plane. J. Combin. Theory Ser. A (1979), 152-160.

Imports (1)

FormalConjecturesUtil

def Erdos214.unitSquare The four vertices of a unit square line 40 source
Source documentation
The four vertices of a unit square.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fin, OfNat.ofNat, Nat, EuclideanSpace, Real
Proof uses: Matrix.vecCons, WithLp.toLp, ENNReal, Matrix.vecEmpty
11 instances, coercions and auxiliary constants left out.
```
def unitSquare : Fin 4 → ℝ² := ![!₂[0, 0], !₂[1, 0], !₂[1, 1], !₂[0, 1]]
```
def Erdos214.UnitDistanceAvoiding Colouring the points of S blue and the points of Sᶜ red gives a red/blue colouring of $\mathbb{R}^2$; it is unit-distance-avoiding if no two blue points are distance $1$ apart line 44 source
Source documentation
Colouring the points of S blue and the points of Sᶜ red gives a red/blue colouring of $\mathbb{R}^2$; it is unit-distance-avoiding if no two blue points are distance $1$ apart.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: Membership.mem, Ne, Dist.dist, ENNReal
14 instances, coercions and auxiliary constants left out.
```
def UnitDistanceAvoiding (S : Set ℝ²) : Prop := ∀ x ∈ S, ∀ y ∈ S, dist x y ≠ 1
```
def Erdos214.HasRedCopies Any unit-distance-avoiding colouring contains a congruent red copy of every set of n points line 48 source
Source documentation
Any unit-distance-avoiding colouring contains a congruent red copy of every set of n points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Erdos214.UnitDistanceAvoiding, Function.Injective, Exists, And, Membership.mem, Compl.compl, Congruent, ENNReal
14 instances, coercions and auxiliary constants left out.
```
def HasRedCopies (n : ℕ) : Prop :=
…
```
def Erdos214.k The largest integer k such that any unit-distance-avoiding colouring contains a congruent red copy of every set of k points line 54 source
Source documentation
The largest integer k such that any unit-distance-avoiding colouring contains a congruent red copy of every set of k points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Erdos214.HasRedCopies
1 instances, coercions and auxiliary constants left out.
```
noncomputable def k : ℕ := sSup {n | HasRedCopies n}
…
```
Mentions: Erdos214.HasRedCopies
theorem Erdos214.erdos_214 Let $S\subset \mathbb{R}^2$ be such that no two points in $S$ are distance $1$ apart sorry line 64 source
Source documentation
Let $S\subset \mathbb{R}^2$ be such that no two points in $S$ are distance $1$ apart. Must the complement of $S$ contain four points which form a unit square?
The answer is yes, proved by Juhász [Ju79], who proved more generally that the complement of $S$ must contain a congruent copy of any set of four points.

depends on sorry what this means
Statement uses: Iff, True, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Erdos214.UnitDistanceAvoiding, Exists, And, Membership.mem, Compl.compl, Congruent, ENNReal, Erdos214.unitSquare
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_214 : answer(True) ↔
    ∀ S : Set ℝ², UnitDistanceAvoiding S →
      ∃ p : Fin 4 → ℝ², (∀ i, p i ∈ Sᶜ) ∧ (p ≅ unitSquare) := by
…
```
Mentions: Erdos214.UnitDistanceAvoiding, Erdos214.unitSquare
theorem Erdos214.erdos_214.variants.juhasz The answer is yes, proved by Juhász [Ju79], who proved more generally that the complement of $S$ must contain a congruent copy of any set of four points sorry line 74 source
Source documentation
The answer is yes, proved by Juhász [Ju79], who proved more generally that the complement of $S$ must contain a congruent copy of any set of four points.

depends on sorry what this means
Statement uses: Erdos214.HasRedCopies, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_214.variants.juhasz : HasRedCopies 4 := by
…
```
Mentions: Erdos214.HasRedCopies
theorem Erdos214.erdos_214.variants.egmrss Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus [EGMRSS75] had earlier showed that $3\leq k\leq 10^{12}$ sorry line 82 source
Source documentation
Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus [EGMRSS75] had earlier showed that $3\leq k\leq 10^{12}$.

depends on sorry what this means
Statement uses: And, LE.le, Nat, OfNat.ofNat, Erdos214.k, HPow.hPow
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_214.variants.egmrss : 3 ≤ k ∧ k ≤ 10 ^ 12 := by
…
```
Mentions: Erdos214.k
theorem Erdos214.erdos_214.variants.juhasz_twelve_points Juhász [Ju79] also improved the upper bound to $k\leq 11$ sorry line 89 source
Source documentation
Juhász [Ju79] also improved the upper bound to $k\leq 11$.

depends on sorry what this means
Statement uses: Not, Erdos214.HasRedCopies, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_214.variants.juhasz_twelve_points : ¬ HasRedCopies 12 := by
…
```
Mentions: Erdos214.HasRedCopies
theorem Erdos214.erdos_214.variants.csizmadia_toth The upper bound was improved further by Csizmadia and Tóth [CsTo94] to $k\leq 7$ sorry line 96 source
Source documentation
The upper bound was improved further by Csizmadia and Tóth [CsTo94] to $k\leq 7$.

depends on sorry what this means
Statement uses: Not, Erdos214.HasRedCopies, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_214.variants.csizmadia_toth : ¬ HasRedCopies 8 := by
…
```
Mentions: Erdos214.HasRedCopies
theorem Erdos214.erdos_214.variants.bounds The best known bounds currently are $$4\leq k\leq 7.$$ sorry line 104 source
Source documentation
The best known bounds currently are $$4\leq k\leq 7.$$

depends on sorry what this means
Statement uses: And, LE.le, Nat, OfNat.ofNat, Erdos214.k
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_214.variants.bounds : 4 ≤ k ∧ k ≤ 7 := by
…
```
Mentions: Erdos214.k