1028.lean`

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

6 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 1028

References:

erdosproblems.com/1028
[Er63d] Erdős, Pál, On combinatorial questions connected with a theorem of Ramsey and van der Waerden. Mat. Lapok (1963), 29-37.
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
[ErSp71] Erdős, P. and Spencer, J., Imbalances in $k$-colorations. Networks (1971/72), 379-385.

Imports (1)

FormalConjecturesUtil

def Erdos1028.imbalance The imbalance of a finite set X under a colouring f, that is, $\left\lvert \sum_{x<y\in X} f(x,y)\right\rvert$ line 41 source
Source documentation
The imbalance of a finite set X under a colouring f, that is, $\left\lvert \sum_{x<y\in X} f(x,y)\right\rvert$.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Int, Finset
Proof uses: abs, Finset.sum, Finset.filter, LT.lt
5 instances, coercions and auxiliary constants left out.
```
def imbalance (f : ℕ → ℕ → ℤ) (X : Finset ℕ) : ℤ :=
…
```
def Erdos1028.H H n is the minimum, over all colourings f of pairs from {1, …, n} with values in {-1, 1}, of the largest imbalance of a subset X ⊆ {1, …, n} line 46 source
Source documentation
H n is the minimum, over all colourings f of pairs from {1, …, n} with values in {-1, 1}, of the largest imbalance of a subset X ⊆ {1, …, n}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Int, And, Or, Eq, OfNat.ofNat, Neg.neg, Finset, LE.le, Finset.Icc, Erdos1028.imbalance
12 instances, coercions and auxiliary constants left out.
```
noncomputable def H (n : ℕ) : ℕ :=
…
```
theorem Erdos1028.erdos_1028 Let $$H(n)=\min_f \max_{X\subseteq \{1,\ldots,n\}} \left\lvert \sum_{x<y\in X} f(x,y)\right\rvert,$$ where $f$ ranges over all functions $f:\{1,\ldots,n\}^2\to \{-1,1\}$ sorry line 60 source
Source documentation
Let $$H(n)=\min_f \max_{X\subseteq \{1,\ldots,n\}} \left\lvert \sum_{x<y\in X} f(x,y)\right\rvert,$$ where $f$ ranges over all functions $f:\{1,\ldots,n\}^2\to \{-1,1\}$. Estimate $H(n)$.
Erdős [Er63d] proved $$\frac{n}{4}\leq H(n) \ll n^{3/2}.$$ Erdős and Spencer [ErSp71] proved that $H(n)\gg n^{3/2}$.

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos1028.H, HPow.hPow, HDiv.hDiv, OfNat.ofNat
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_1028 :
    (fun n => (H n : ℝ)) =Θ[atTop]
      fun n : ℕ => (n : ℝ) ^ (3 / 2 : ℝ) := by
…
```
Mentions: Erdos1028.H
theorem Erdos1028.erdos_1028.variants.lower_bound Erdős [Er63d] proved $$\frac{n}{4}\leq H(n) \ll n^{3/2}.$$ sorry line 69 source
Source documentation
Erdős [Er63d] proved $$\frac{n}{4}\leq H(n) \ll n^{3/2}.$$

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, LE.le, Real, HDiv.hDiv, OfNat.ofNat, Erdos1028.H, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1028.variants.lower_bound :
    ∀ᶠ n : ℕ in atTop, (n : ℝ) / 4 ≤ (H n : ℝ) := by
…
```
Mentions: Erdos1028.H
theorem Erdos1028.erdos_1028.variants.upper_bound Erdős [Er63d] proved $$\frac{n}{4}\leq H(n) \ll n^{3/2}.$$ sorry line 77 source
Source documentation
Erdős [Er63d] proved $$\frac{n}{4}\leq H(n) \ll n^{3/2}.$$

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos1028.H, HPow.hPow, HDiv.hDiv, OfNat.ofNat
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_1028.variants.upper_bound :
    (fun n => (H n : ℝ)) ≪ (fun n : ℕ => (n : ℝ) ^ (3 / 2 : ℝ)) := by
…
```
Mentions: Erdos1028.H
theorem Erdos1028.erdos_1028.variants.erdos_spencer Erdős and Spencer [ErSp71] proved that $H(n)\gg n^{3/2}$ sorry line 83 source
Source documentation
Erdős and Spencer [ErSp71] proved that $H(n)\gg n^{3/2}$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HPow.hPow, HDiv.hDiv, OfNat.ofNat, Erdos1028.H
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_1028.variants.erdos_spencer :
    (fun n : ℕ => (n : ℝ) ^ (3 / 2 : ℝ)) ≪ (fun n => (H n : ℝ)) := by
…
```
Mentions: Erdos1028.H