535.lean`

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

10 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 535

References:

erdosproblems.com/535
[Er64] P. Erdős, On a problem in elementary number theory and a combinatorial problem. Math. Comp. (1964), 644–646.
[AbHa70] H. L. Abbott and D. Hanson, An extremal problem in number theory. Bull. London Math. Soc. (1970), 324–326.
[Er73] P. Erdős, Problems and results on combinatorial number theory, in A Survey of Combinatorial Theory, North-Holland, 1973.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def Erdos535.NoConstantPairwiseGcdCoprimeSubsets No r-subset has constant pairwise GCD with coprime quotients line 42 source
Source documentation
No r-subset has constant pairwise GCD with coprime quotients.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: LE.le, Eq, Finset.card, Not, Exists, And, LT.lt, OfNat.ofNat, Set.Pairwise, Nat.gcd, Membership.mem, HMul.hMul
10 instances, coercions and auxiliary constants left out.
```
def NoConstantPairwiseGcdCoprimeSubsets (r : ℕ) (A : Finset ℕ) : Prop :=
…
```
def Erdos535.AllBigOmega All elements of A are positive and have exactly k prime factors, counted with multiplicity line 55 source
Source documentation
All elements of A are positive and have exactly k prime factors, counted with multiplicity.
Erdős [Er73] explains that Abbott pointed out the ordinary sunflower conjecture does not seem to suffice for Problem 535; the corrected stronger auxiliary statement uses $Ω$, not $ω$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Membership.mem, And, LE.le, OfNat.ofNat, Eq, ArithmeticFunction, ArithmeticFunction.cardFactors
8 instances, coercions and auxiliary constants left out.
```
def AllBigOmega (k : ℕ) (A : Finset ℕ) : Prop :=
…
```
def Erdos535.f f r N is the maximum size of a subset A ⊆ {1,…,N} such that no r-element subset of A has constant pairwise GCD line 60 source
Source documentation
f r N is the maximum size of a subset A ⊆ {1,…,N} such that no r-element subset of A has constant pairwise GCD.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Eq, Finset.card, Not, Set.Pairwise, Nat.gcd
9 instances, coercions and auxiliary constants left out.
```
noncomputable def f (r N : ℕ) : ℕ :=
…
```
theorem Erdos535.erdos_535 Documented declaration: erdos_535 sorry line 76 source
Source documentation
Let $r \geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that $f_3(N) > N^{c/\log\log N}$ for some constant $c > 0$, and conjectured this should also be an upper bound; here we state the conjectural upper bound for all $r \geq 3$.
See also [536].

depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Exists, Real, And, GT.gt, Filter.Eventually, LE.le, Erdos535.f, HPow.hPow, HDiv.hDiv, Real.log, Filter.atTop
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_535 : ∀ r ≥ 3, ∃ c > (0 : ℝ),
    ∀ᶠ (N : ℕ) in atTop,
      (f r N : ℝ) ≤ (N : ℝ) ^ (c / log (log (N : ℝ))) := by
…
```
Mentions: c, Erdos535.f
theorem Erdos535.erdos_535.variants.first_open_case The first open case of Erdős Problem 535 is $r = 3$: there should exist $c > 0$ such that $f_3(N) \leq N^{c/\log\log N}$ for all sufficiently large $N$ sorry line 86 source
Source documentation
The first open case of Erdős Problem 535 is $r = 3$: there should exist $c > 0$ such that $f_3(N) \leq N^{c/\log\log N}$ for all sufficiently large $N$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Erdos535.f, HPow.hPow, HDiv.hDiv, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_535.variants.first_open_case : ∃ c > (0 : ℝ),
    ∀ᶠ (N : ℕ) in atTop,
      (f 3 N : ℝ) ≤ (N : ℝ) ^ (c / log (log (N : ℝ))) := by
…
```
Mentions: c, Erdos535.f
theorem Erdos535.erdos_535.variants.erdos_upper_bound Erdős [Er64] proved that $f_r(N) \leq N^{3/4+o(1)}$ for all $r \geq 3$ sorry line 93 source
Source documentation
Erdős [Er64] proved that $f_r(N) \leq N^{3/4+o(1)}$ for all $r \geq 3$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Real, GT.gt, Filter.Eventually, Erdos535.f, HPow.hPow, HAdd.hAdd, HDiv.hDiv, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_535.variants.erdos_upper_bound {r : ℕ} (hr : 3 ≤ r) :
    ∀ ε > (0 : ℝ), ∀ᶠ (N : ℕ) in atTop,
      (f r N : ℝ) ≤ (N : ℝ) ^ ((3 : ℝ) / 4 + ε) := by
…
```
Mentions: Erdos535.f
theorem Erdos535.erdos_535.variants.lower_bound Erdős [Er64] proved that $f_3(N) > N^{c/\log\log N}$ for some constant $c > 0$ sorry line 100 source
Source documentation
Erdős [Er64] proved that $f_3(N) > N^{c/\log\log N}$ for some constant $c > 0$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, HPow.hPow, HDiv.hDiv, Real.log, Erdos535.f, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_535.variants.lower_bound : ∃ c > (0 : ℝ),
    ∀ᶠ (N : ℕ) in atTop,
      (N : ℝ) ^ (c / log (log (N : ℝ))) ≤ (f 3 N : ℝ) := by
…
```
Mentions: c, Erdos535.f
theorem Erdos535.erdos_535.variants.abbott_hanson Abbott and Hanson [AbHa70] improved Erdős's upper bound to $f_r(N) \leq N^{1/2+o(1)}$ for all $r \geq 3$ sorry line 108 source
Source documentation
Abbott and Hanson [AbHa70] improved Erdős's upper bound to $f_r(N) \leq N^{1/2+o(1)}$ for all $r \geq 3$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Real, GT.gt, Filter.Eventually, Erdos535.f, HPow.hPow, HAdd.hAdd, HDiv.hDiv, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_535.variants.abbott_hanson {r : ℕ} (hr : 3 ≤ r) :
    ∀ ε > (0 : ℝ), ∀ᶠ (N : ℕ) in atTop,
      (f r N : ℝ) ≤ (N : ℝ) ^ ((1 : ℝ) / 2 + ε) := by
…
```
Mentions: Erdos535.f
theorem Erdos535.erdos_535.variants.sunflower_strong Erdős [Er73] records that Abbott pointed out the ordinary sunflower conjecture does not seem to suffice here sorry line 120 source
Source documentation
Erdős [Er73] records that Abbott pointed out the ordinary sunflower conjecture does not seem to suffice here. The stronger auxiliary conjecture uses $Ω(n)=k$, i.e. prime factors counted with multiplicity; this stronger statement would imply the conjectured upper bound for $f_r(N)$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Real, And, GT.gt, Finset, Erdos535.AllBigOmega, Erdos535.NoConstantPairwiseGcdCoprimeSubsets, Finset.card, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_535.variants.sunflower_strong {r : ℕ} (hr : 3 ≤ r) :
    ∃ c_r > (0 : ℝ),
      ∀ k : ℕ, ∀ A : Finset ℕ,
        AllBigOmega k A →
        NoConstantPairwiseGcdCoprimeSubsets r A →
        (A.card : ℝ) ≤ c_r ^ k := by
…
```
Mentions: Erdos535.AllBigOmega, Erdos535.NoConstantPairwiseGcdCoprimeSubsets
theorem Erdos535.erdos_535.variants.sunflower_erdos_rado For the stronger $Ω(n)=k$ variant above, the Erdős–Rado method gives the weaker bound $c_r^k \cdot k!$; see Erdős [Er73] sorry line 133 source
Source documentation
For the stronger $Ω(n)=k$ variant above, the Erdős–Rado method gives the weaker bound $c_r^k \cdot k!$; see Erdős [Er73].

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Real, And, GT.gt, Finset, Erdos535.AllBigOmega, Erdos535.NoConstantPairwiseGcdCoprimeSubsets, Finset.card, HMul.hMul, HPow.hPow, Nat.factorial
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_535.variants.sunflower_erdos_rado {r : ℕ} (hr : 3 ≤ r) :
    ∃ c_r > (0 : ℝ),
      ∀ k : ℕ, ∀ A : Finset ℕ,
        AllBigOmega k A →
        NoConstantPairwiseGcdCoprimeSubsets r A →
        (A.card : ℝ) ≤ c_r ^ k * (Nat.factorial k) := by
…
```
Mentions: Erdos535.AllBigOmega, Erdos535.NoConstantPairwiseGcdCoprimeSubsets