798.lean`

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

6 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 798

References:

erdosproblems.com/798
[Al91] Alon, N., Economical coverings of sets of lattice points. Geom. Funct. Anal. (1991), 224-230.

Imports (1)

FormalConjecturesUtil

def Erdos798.gridPoints The grid $\{1,\ldots,n\}^2$, viewed as a set of points of the plane line 38 source
Source documentation
The grid $\{1,\ldots,n\}^2$, viewed as a set of points of the plane.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Prod, Real
Proof uses: Set.ofPred, Exists, And, LE.le, OfNat.ofNat, Eq, Prod.mk
4 instances, coercions and auxiliary constants left out.
```
def gridPoints (n : ℕ) : Set (ℝ × ℝ) :=
…
```
def Erdos798.IsLineCover S is a set of points of $\{1,\ldots,n\}^2$ such that the lines determined by the pairs of distinct points of S cover all points of $\{1,\ldots,n\}^2$ line 45 source
Source documentation
S is a set of points of $\{1,\ldots,n\}^2$ such that the lines determined by the pairs of distinct points of S cover all points of $\{1,\ldots,n\}^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Prod, Real
Proof uses: And, LE.le, Erdos798.gridPoints, Membership.mem, Exists, Ne, Collinear, Insert.insert, Singleton.singleton
23 instances, coercions and auxiliary constants left out.
```
def IsLineCover (n : ℕ) (S : Set (ℝ × ℝ)) : Prop :=
…
```
def Erdos798.t The minimum number of points in $\{1,\ldots,n\}^2$ such that the lines determined by these points cover all points in $\{1,\ldots,n\}^2$ line 53 source
Source documentation
The minimum number of points in $\{1,\ldots,n\}^2$ such that the lines determined by these points cover all points in $\{1,\ldots,n\}^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Set, Prod, Real, And, Erdos798.IsLineCover, Eq, Set.ncard
1 instances, coercions and auxiliary constants left out.
```
noncomputable def t (n : ℕ) : ℕ :=
…
```
theorem Erdos798.erdos_798 Let $t(n)$ be the minimum number of points in $\{1,\ldots,n\}^2$ such that the $\binom{t}{2}$ lines determined by these points cover all points in $\{1,\ldots,n\}^2$ sorry line 67 source
Source documentation
Let $t(n)$ be the minimum number of points in $\{1,\ldots,n\}^2$ such that the $\binom{t}{2}$ lines determined by these points cover all points in $\{1,\ldots,n\}^2$.
Estimate $t(n)$. In particular, is it true that $t(n)=o(n)$?
A problem of Erdős and Purdy, who proved $t(n) \gg n^{2/3}$.
Resolved by Alon [Al91] who proved $t(n) \ll n^{2/3}\log n$.

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos798.t
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_798 : answer(True) ↔
    (fun n : ℕ => (t n : ℝ)) =o[atTop] (fun n : ℕ => (n : ℝ)) := by
…
```
Mentions: Erdos798.t
theorem Erdos798.erdos_798.variants.lower_bound A problem of Erdős and Purdy, who proved $t(n) \gg n^{2/3}$ sorry line 75 source
Source documentation
A problem of Erdős and Purdy, who proved $t(n) \gg n^{2/3}$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HPow.hPow, HDiv.hDiv, OfNat.ofNat, Erdos798.t
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_798.variants.lower_bound :
    (fun n : ℕ => (n : ℝ) ^ (2 / 3 : ℝ)) =O[atTop] (fun n : ℕ => (t n : ℝ)) := by
…
```
Mentions: Erdos798.t
theorem Erdos798.erdos_798.variants.upper_bound Resolved by Alon [Al91] who proved $t(n) \ll n^{2/3}\log n$ sorry line 83 source
Source documentation
Resolved by Alon [Al91] who proved $t(n) \ll n^{2/3}\log n$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos798.t, HMul.hMul, HPow.hPow, HDiv.hDiv, OfNat.ofNat, Real.log
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_798.variants.upper_bound :
    (fun n : ℕ => (t n : ℝ)) =O[atTop]
      (fun n : ℕ => (n : ℝ) ^ (2 / 3 : ℝ) * Real.log n) := by
…
```
Mentions: Erdos798.t