501.lean`

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

13 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 501

References:

erdosproblems.com/501
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221-254.
[ErHa71] Erdős, Paul and Hajnal, András, Unsolved problems in set theory. Axiomatic Set Theory, Proc. Sympos. Pure Math. XIII Part I (1971), 17-48.
[ErHa60] Erdős, Paul and Hajnal, András. On some combinatorial problems involving complete graphs. Acta Math. Acad. Sci. Hungar. (1960), 395-424.
[Gl62] Gladysz, S. Some topological properties of independent sets. Colloq. Math. (1962).
[He72] Hechler, S. H. A dozen small uncountable cardinals. TOPO 72, Lecture Notes in Math. (1972), 207-218.
[NPS87] Newelski, L., Pawlikowski, J., and Seredyński, F. Infinite independent sets in the closed case. Acta Math. Acad. Sci. Hungar. (1987).

Imports (1)

FormalConjecturesUtil

theorem Erdos501.erdos_501 For every $x \in \mathbb{R}$ let $A_x \subset \mathbb{R}$ be a bounded set with outer measure $< 1$ sorry line 71 source
Source documentation
For every $x \in \mathbb{R}$ let $A_x \subset \mathbb{R}$ be a bounded set with outer measure $< 1$. Must there exist an infinite independent set, that is, some infinite $X \subseteq \mathbb{R}$ such that $x \notin A_y$ for all $x \neq y \in X$?
If the sets $A_x$ are closed and have measure $< 1$, then must there exist an independent set of size $3$?
Known results: Erdős–Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets. Hechler [He72] showed the answer is no assuming the continuum hypothesis.

depends on sorry what this means
Statement uses: Iff, True, Real, Set, Bornology.IsBounded, LT.lt, ENNReal, MeasureTheory.OuterMeasure, MeasureTheory.Measure.toOuterMeasure, MeasureTheory.MeasureSpace.volume, OfNat.ofNat, Exists, And, Set.Infinite, Set.Pairwise, Not, Membership.mem
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_501 : answer(sorry) ↔
    ∀ (A : ℝ → Set ℝ),
      (∀ x, Bornology.IsBounded (A x)) →
      (∀ x, volume.toOuterMeasure (A x) < 1) →
      ∃ X : Set ℝ, X.Infinite ∧ X.Pairwise (fun x y => x ∉ A y) := by
…
```
theorem Erdos501.erdos_501.variants.erdosHajnal_finite Erdős–Hajnal (1960): arbitrarily large finite independent sets exist sorry line 88 source
Source documentation
Erdős–Hajnal (1960): arbitrarily large finite independent sets exist.
For every n : ℕ and every family A : ℝ → Set ℝ of bounded sets with Lebesgue outer measure < 1, there exists a finite independent set of size at least n.
This was proved by Erdős and Hajnal [ErHa60].

depends on sorry what this means
Statement uses: Iff, True, Nat, Real, Set, Bornology.IsBounded, LT.lt, ENNReal, MeasureTheory.OuterMeasure, MeasureTheory.Measure.toOuterMeasure, MeasureTheory.MeasureSpace.volume, OfNat.ofNat, Exists, Finset, And, LE.le, Finset.card, Set.Pairwise, Not, Membership.mem
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_501.variants.erdosHajnal_finite : answer(True) ↔
    ∀ (n : ℕ) (A : ℝ → Set ℝ),
      (∀ x, Bornology.IsBounded (A x)) →
      (∀ x, volume.toOuterMeasure (A x) < 1) →
      ∃ X : Finset ℝ, n ≤ X.card ∧ (X : Set ℝ).Pairwise (fun x y => x ∉ A y) := by
…
```
theorem Erdos501.erdos_501.variants.hechler_CH Hechler (1972) [He72]: the answer to the main question is NO, assuming the continuum hypothesis sorry line 102 source
Source documentation
Hechler (1972) [He72]: the answer to the main question is NO, assuming the continuum hypothesis.
Assuming CH (ℵ₁ = 𝔠), there exists a family A : ℝ → Set ℝ of bounded sets with Lebesgue outer measure < 1 for which no infinite independent set exists.

depends on sorry what this means
Statement uses: Iff, True, Eq, Cardinal, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Cardinal.continuum, Exists, Real, Set, And, Bornology.IsBounded, LT.lt, ENNReal, MeasureTheory.OuterMeasure, MeasureTheory.Measure.toOuterMeasure, MeasureTheory.MeasureSpace.volume, Not and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
35 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_501.variants.hechler_CH : answer(True) ↔
    (ℵ₁ = 𝔠) →
    ∃ (A : ℝ → Set ℝ),
      (∀ x, Bornology.IsBounded (A x)) ∧
      (∀ x, volume.toOuterMeasure (A x) < 1) ∧
      ¬ ∃ X : Set ℝ, X.Infinite ∧ X.Pairwise (fun x y => x ∉ A y) := by
…
```
theorem Erdos501.erdos_501.variants.closed_size3 sorry line 120 source
depends on sorry what this means
Statement uses: Iff, True, Real, Set, IsClosed, LT.lt, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, OfNat.ofNat, Exists, And, LE.le, Nat, Set.ncard, Set.Pairwise, Not, Membership.mem
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos501.erdos_501.variants.newelski_pawlikowski_seredynski sorry line 135 source
depends on sorry what this means
Statement uses: Iff, True, Real, Set, IsClosed, LT.lt, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, OfNat.ofNat, Exists, And, Set.Infinite, Set.Pairwise, Not, Membership.mem
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos501.erdos_501.variants.gladysz_size2 Gladysz (1962) [Gl62]: independent set of size 2 in the closed case sorry line 151 source
Source documentation
Gladysz (1962) [Gl62]: independent set of size 2 in the closed case.
If all the sets A x are closed with Lebesgue measure < 1, then there exist two distinct reals x y such that x ∉ A y and y ∉ A x.
This is a weaker result proved by Gladysz before the full Newelski–Pawlikowski– Seredyński theorem [NPS87].

depends on sorry what this means
Statement uses: Iff, True, Real, Set, IsClosed, LT.lt, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, OfNat.ofNat, Exists, And, LE.le, Nat, Set.ncard, Set.Pairwise, Not, Membership.mem
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_501.variants.gladysz_size2 : answer(True) ↔
    ∀ (A : ℝ → Set ℝ),
      (∀ x, IsClosed (A x)) →
      (∀ x, volume (A x) < 1) →
      ∃ X : Set ℝ, 2 ≤ X.ncard ∧ X.Pairwise (fun x y => x ∉ A y) := by
…
```
theorem Erdos501.erdos_501.variants.singleton_independent Trivial lower bound: a single-element set is always independent line 164 source
Source documentation
Trivial lower bound: a single-element set is always independent.
For any family A, any singleton {x} is vacuously independent: there are no two distinct elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set, Set.Pairwise, Singleton.singleton, Not, Membership.mem
Proof uses: Set.pairwise_singleton
3 instances, coercions and auxiliary constants left out.
```
theorem erdos_501.variants.singleton_independent (A : ℝ → Set ℝ) (x : ℝ) :
    ({x} : Set ℝ).Pairwise (fun x y => x ∉ A y) :=
…
```
theorem Erdos501.erdos_501.variants.pair_independent_iff Two-element sets: independent iff mutual non-membership line 174 source
Source documentation
Two-element sets: independent iff mutual non-membership.
A two-element set {x, y} (with x ≠ y) is independent for A if and only if x ∉ A y and y ∉ A x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set, Ne, Iff, Set.Pairwise, Insert.insert, Singleton.singleton, Not, Membership.mem, And
Proof uses: Eq.mpr, id, Eq, congrArg, propext, Set.pairwise_pair, Iff.intro
6 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_501.variants.pair_independent_iff (A : ℝ → Set ℝ) {x y : ℝ} (hxy : x ≠ y) :
    ({x, y} : Set ℝ).Pairwise (fun x y => x ∉ A y) ↔ x ∉ A y ∧ y ∉ A x := by
…
```
theorem Erdos501.erdos_501.tests.empty_family_is_valid The constant family A x = ∅ satisfies all hypotheses of the main problem: each A x is bounded (the empty set is bounded) and has Lebesgue outer measure 0 < 1 line 191 source
Source documentation
The constant family A x = ∅ satisfies all hypotheses of the main problem: each A x is bounded (the empty set is bounded) and has Lebesgue outer measure 0 < 1. Moreover, all of ℝ is an independent set, showing the conclusion holds trivially.
This demonstrates that the hypotheses are non-vacuous: the family A x = ∅ is a valid input to the theorem, and ℝ (which is infinite) witnesses the conclusion.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set, EmptyCollection.emptyCollection, And, Bornology.IsBounded, LT.lt, ENNReal, MeasureTheory.OuterMeasure, MeasureTheory.Measure.toOuterMeasure, MeasureTheory.MeasureSpace.volume, OfNat.ofNat, Exists, Set.Infinite, Set.Pairwise, Not, Membership.mem
Proof uses: And.intro, Bornology.isBounded_empty, of_eq_true, Eq.trans, True, congrFun', congrArg, MeasureTheory.measure_empty, Exists.intro, Set.univ, Set.infinite_univ, FirstOrder.Language.order, Ne
66 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_501.tests.empty_family_is_valid :
    let A : ℝ → Set ℝ := fun _ => ∅
…
```
theorem Erdos501.erdos_501.tests.singleton_zero_independent A singleton {0} is an independent set for any family A : ℝ → Set ℝ, as witnessed by erdos_501.variants.singleton_independent line 203 source
Source documentation
A singleton {0} is an independent set for any family A : ℝ → Set ℝ, as witnessed by erdos_501.variants.singleton_independent.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set, Set.Pairwise, Singleton.singleton, OfNat.ofNat, Not, Membership.mem
Proof uses: Erdos501.erdos_501.variants.singleton_independent
6 instances, coercions and auxiliary constants left out.
```
theorem erdos_501.tests.singleton_zero_independent (A : ℝ → Set ℝ) :
    ({0} : Set ℝ).Pairwise (fun x y => x ∉ A y) :=
…
```
theorem Erdos501.erdos_501.tests.pair_independent_empty Two reals form an independent set for the empty family A _ = ∅: neither 0 nor 1 belongs to ∅, so both conditions of pair_independent_iff hold line 210 source
Source documentation
Two reals form an independent set for the empty family A _ = ∅: neither 0 nor 1 belongs to ∅, so both conditions of pair_independent_iff hold.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set.Pairwise, Real, Insert.insert, Set, OfNat.ofNat, Singleton.singleton, Not, Membership.mem, EmptyCollection.emptyCollection
Proof uses: Eq.mpr, And, id, Eq, congrArg, propext, Erdos501.erdos_501.variants.pair_independent_iff, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Nat.cast_one, Eq.refl, Bool, Bool.false, And.intro, Set.notMem_empty
25 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_501.tests.pair_independent_empty :
    ({0, 1} : Set ℝ).Pairwise (fun x y => x ∉ (fun _ => (∅ : Set ℝ)) y) := by
…
```
theorem Erdos501.erdos_501.tests.singleton_outer_measure_lt_one The hypothesis volume.toOuterMeasure (A x) < 1 is strictly satisfied when A x = {x} (a singleton), since Lebesgue measure of a singleton is 0 line 218 source
Source documentation
The hypothesis volume.toOuterMeasure (A x) < 1 is strictly satisfied when A x = {x} (a singleton), since Lebesgue measure of a singleton is 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LT.lt, ENNReal, MeasureTheory.OuterMeasure, Set, MeasureTheory.Measure.toOuterMeasure, MeasureTheory.MeasureSpace.volume, Singleton.singleton, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, MeasureTheory.NullSingletonClass.measure_singleton
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_501.tests.singleton_outer_measure_lt_one (x : ℝ) :
    volume.toOuterMeasure ({x} : Set ℝ) < 1 := by
…
```
theorem Erdos501.erdos_501.tests.unit_interval_measure The boundary case: the measure condition < 1 is sharp line 225 source
Source documentation
The boundary case: the measure condition < 1 is sharp. An interval of length ≥ 1 has Lebesgue measure ≥ 1, so it would fail the hypothesis. Here [0, 1] has measure exactly 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, ENNReal, MeasureTheory.OuterMeasure, Real, Set, MeasureTheory.Measure.toOuterMeasure, MeasureTheory.MeasureSpace.volume, Set.Icc, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, MeasureTheory.Measure, ENNReal.ofReal, HSub.hSub, Real.volume_Icc, sub_zero, ENNReal.ofReal_one, eq_self
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_501.tests.unit_interval_measure :
    volume.toOuterMeasure (Set.Icc (0:ℝ) 1) = 1 := by
…
```