813.lean`

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

9 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 813

References:

erdosproblems.com/813
[Er91] Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.
[BuSu23] Bucić, M. and Sudakov, B., Large independent sets from local considerations. Combinatorica 43 (2023), 505-546. arXiv:2007.03667

Imports (1)

FormalConjecturesUtil

def Erdos813.EverySevenHasTriangle Every set of $7$ vertices of G contains a triangle, that is, three pairwise adjacent vertices line 42 source
Source documentation
Every set of $7$ vertices of G contains a triangle, that is, three pairwise adjacent vertices. For $n < 7$ there is no such set, so the condition is vacuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph, Fin
Proof uses: Finset, Eq, Finset.card, OfNat.ofNat, Exists, And, LE.le, SimpleGraph.IsNClique
4 instances, coercions and auxiliary constants left out.
```
def EverySevenHasTriangle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
…
```
def Erdos813.h h n is $h(n)$: the smallest clique number of a graph on $n$ vertices in which every set of $7$ vertices contains a triangle line 50 source
Source documentation
h n is $h(n)$: the smallest clique number of a graph on $n$ vertices in which every set of $7$ vertices contains a triangle. The complete graph satisfies the condition, so the set is never empty and sInf is a true minimum.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, SimpleGraph, Fin, And, Erdos813.EverySevenHasTriangle, Eq, SimpleGraph.cliqueNum
1 instances, coercions and auxiliary constants left out.
```
noncomputable def h (n : ℕ) : ℕ :=
…
```
theorem Erdos813.everySevenHasTriangle_top The complete graph on $n$ vertices satisfies the condition line 55 source
Source documentation
The complete graph on $n$ vertices satisfies the condition.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Erdos813.EverySevenHasTriangle, Top.top, SimpleGraph, Fin
Proof uses: Finset, Eq, Finset.card, OfNat.ofNat, And, LE.le, Exists, SimpleGraph.IsNClique, Finset.exists_subset_card_eq, Decidable.byContradiction, Not, Exists.intro, And.intro, SimpleGraph.IsNClique.mk, SimpleGraph.IsClique.subset, Set.univ, Set.subset_univ, Iff.mpr, SimpleGraph.IsClique, SimpleGraph.isClique_univ and 1 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem everySevenHasTriangle_top (n : ℕ) :
    EverySevenHasTriangle (⊤ : SimpleGraph (Fin n)) := by
…
```
Mentions: Erdos813.EverySevenHasTriangle
theorem Erdos813.h_seven For $n = 7$ the graph must contain a triangle, and a triangle plus $4$ isolated vertices has clique number $3$, so $h(7) = 3$ sorry line 64 source
Source documentation
For $n = 7$ the graph must contain a triangle, and a triangle plus $4$ isolated vertices has clique number $3$, so $h(7) = 3$.

depends on sorry what this means
Statement uses: Eq, Nat, Erdos813.h, OfNat.ofNat
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 h_seven : h 7 = 3 := by
…
```
Mentions: Erdos813.h
theorem Erdos813.erdos_813 Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle must contain a clique on at least $h(n)$ vertices sorry line 77 source
Source documentation
Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle must contain a clique on at least $h(n)$ vertices. Estimate $h(n)$. In particular, do there exist constants $c_1, c_2 > 0$ such that $$n^{1/3+c_1} \ll h(n) \ll n^{1/2-c_2}?$$
The lower half has a positive answer (Erdos813.erdos_813.variants.lower); the upper half is Erdos813.erdos_813.variants.upper.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, HPow.hPow, HAdd.hAdd, HDiv.hDiv, Erdos813.h, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_813 : answer(sorry) ↔
    ∃ c₁ > (0 : ℝ), ∃ c₂ > (0 : ℝ),
      (fun n : ℕ ↦ (n : ℝ) ^ (1 / 3 + c₁ : ℝ)) =O[atTop] (fun n ↦ (h n : ℝ)) ∧
        (fun n : ℕ ↦ (h n : ℝ)) =O[atTop] (fun n : ℕ ↦ (n : ℝ) ^ (1 / 2 - c₂ : ℝ)) := by
…
```
Mentions: Erdos813.h
theorem Erdos813.erdos_813.variants.erdos_hajnal Erdős and Hajnal proved (see [Er91]) that $$n^{1/3} \ll h(n) \ll n^{1/2}.$$ sorry line 88 source
Source documentation
Erdős and Hajnal proved (see [Er91]) that $$n^{1/3} \ll h(n) \ll n^{1/2}.$$

depends on sorry what this means
Statement uses: And, Asymptotics.IsBigO, Nat, Real, Filter.atTop, HPow.hPow, HDiv.hDiv, OfNat.ofNat, Erdos813.h
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_813.variants.erdos_hajnal :
    (fun n : ℕ ↦ (n : ℝ) ^ (1 / 3 : ℝ)) =O[atTop] (fun n ↦ (h n : ℝ)) ∧
      (fun n : ℕ ↦ (h n : ℝ)) =O[atTop] (fun n : ℕ ↦ (n : ℝ) ^ (1 / 2 : ℝ)) := by
…
```
Mentions: Erdos813.h
theorem Erdos813.erdos_813.variants.bucic_sudakov Bucić and Sudakov [BuSu23] proved that $h(n) \gg n^{5/12-o(1)}$ sorry line 99 source
Source documentation
Bucić and Sudakov [BuSu23] proved that $h(n) \gg n^{5/12-o(1)}$. Their result is stated for graphs in which every $7$ vertices contain an independent set of size $3$; it applies to $h$ by taking complements.

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, HPow.hPow, HSub.hSub, HDiv.hDiv, Erdos813.h, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_813.variants.bucic_sudakov :
    ∀ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop, (n : ℝ) ^ (5 / 12 - ε : ℝ) ≤ (h n : ℝ) := by
…
```
Mentions: Erdos813.h
theorem Erdos813.erdos_813.variants.lower The lower half of Erdos813.erdos_813 has a positive answer: there is a constant $c_1 > 0$ with $n^{1/3+c_1} \ll h(n)$ sorry line 109 source
Source documentation
The lower half of Erdos813.erdos_813 has a positive answer: there is a constant $c_1 > 0$ with $n^{1/3+c_1} \ll h(n)$. This follows from the bound $h(n) \gg n^{5/12-o(1)}$ of Bucić and Sudakov [BuSu23], since $5/12 > 1/3$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, HPow.hPow, HAdd.hAdd, HDiv.hDiv, Erdos813.h
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_813.variants.lower : answer(True) ↔
    ∃ c₁ > (0 : ℝ), (fun n : ℕ ↦ (n : ℝ) ^ (1 / 3 + c₁ : ℝ)) =O[atTop] (fun n ↦ (h n : ℝ)) := by
…
```
Mentions: Erdos813.h
theorem Erdos813.erdos_813.variants.upper The upper half of Erdos813.erdos_813: is there a constant $c_2 > 0$ with $h(n) \ll n^{1/2-c_2}$? sorry line 118 source
Source documentation
The upper half of Erdos813.erdos_813: is there a constant $c_2 > 0$ with $h(n) \ll n^{1/2-c_2}$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, Erdos813.h, HPow.hPow, HSub.hSub, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_813.variants.upper : answer(sorry) ↔
    ∃ c₂ > (0 : ℝ), (fun n : ℕ ↦ (h n : ℝ)) =O[atTop] (fun n : ℕ ↦ (n : ℝ) ^ (1 / 2 - c₂ : ℝ)) := by
…
```
Mentions: Erdos813.h