920.lean`

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

6 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 920

References:

erdosproblems.com/166
erdosproblems.com/920
erdosproblems.com/986
erdosproblems.com/1104
[Br26] D. Brada\v{c}, Nearly tight exponents for off-diagonal Ramsey numbers. arXiv:2605.28793 (2026).
[Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.
[GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory (1968), 125--175.
[MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos920.f $f_k(n)$ is the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$ line 47 source
Source documentation
$f_k(n)$ is the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, SimpleGraph, Fin, SimpleGraph.CliqueFree, Eq, ENat, SimpleGraph.chromaticNumber
3 instances, coercions and auxiliary constants left out.
```
noncomputable def f (k n : ℕ) : ℕ :=
…
```
theorem Erdos920.erdos_920 sorry line 59 source
depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Exists, And, GT.gt, Asymptotics.IsBigO, Real, Filter.atTop, HDiv.hDiv, HPow.hPow, HSub.hSub, Real.log, Erdos920.f
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 Erdos920.erdos_920.variants.upper_bound Graver and Yackel [GrYa68] proved that $f_k(n) \ll \left(n\frac{\log\log n}{\log n}\right)^{1-\frac{1}{k-1}}.$ sorry line 69 source
Source documentation
Graver and Yackel [GrYa68] proved that $f_k(n) \ll \left(n\frac{\log\log n}{\log n}\right)^{1-\frac{1}{k-1}}.$

depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Asymptotics.IsBigO, Real, Filter.atTop, Erdos920.f, HPow.hPow, HDiv.hDiv, HMul.hMul, Real.log, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_920.variants.upper_bound (k : ℕ) (hk : k ≥ 3) :
    (fun n ↦ f k n) ≪ (fun n ↦ ((n : ℝ) * log (log n) / log n) ^ (1 - 1 / ((k : ℝ) - 1))) := by
…
```
Mentions: Erdos920.f
theorem Erdos920.erdos_920.variants.k_eq_3 It is known that $f_3(n)\asymp (n/\log n)^{1/2}$ (see [erdosproblems.com/1104]) sorry line 77 source
Source documentation
It is known that $f_3(n)\asymp (n/\log n)^{1/2}$ (see [erdosproblems.com/1104]).

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos920.f, OfNat.ofNat, HPow.hPow, HDiv.hDiv, 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_920.variants.k_eq_3 :
    (fun n ↦ (f 3 n : ℝ)) =Θ[atTop] (fun n ↦ ((n : ℝ) / log n) ^ (1 / 2 : ℝ)) := by
…
```
Mentions: Erdos920.f
theorem Erdos920.erdos_920.variants.lower_bound_k_eq_4 The lower bound $R(4,m) \gg m^3/(\log m)^4$ of Mattheus and Verstraete [MaVe23] (see [erdosproblems.com/166]) implies $f_4(n) \gg \frac{n^{2/3}}{(\log n)^{4/3}}$ sorry line 86 source
Source documentation
The lower bound $R(4,m) \gg m^3/(\log m)^4$ of Mattheus and Verstraete [MaVe23] (see [erdosproblems.com/166]) implies $f_4(n) \gg \frac{n^{2/3}}{(\log n)^{4/3}}$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HDiv.hDiv, HPow.hPow, OfNat.ofNat, Real.log, Erdos920.f
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_920.variants.lower_bound_k_eq_4 :
    (fun n ↦ f 4 n) ≫ (fun n ↦ (n : ℝ) ^ (2 / 3 : ℝ) / (log n) ^ (4 / 3 : ℝ)) := by
…
```
Mentions: Erdos920.f
theorem Erdos920.erdos_920.variants.lower_bound_k_ge_5 A positive answer to this question for all sorry line 96 source
Source documentation
A positive answer to this question for all $k\geq 5$, i.e. $f_k(n) \gg \frac{n^{1-\frac{1}{k-1}}}{(\log n)^{c_k}}$ for some constant $c_k>0$, follows from the lower bound in [erdosproblems.com/986] given by Bradač [Br26].

depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Exists, And, GT.gt, Asymptotics.IsBigO, Real, Filter.atTop, HDiv.hDiv, HPow.hPow, HSub.hSub, Real.log, Erdos920.f
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_920.variants.lower_bound_k_ge_5 (k : ℕ) (hk : k ≥ 5) :
    ∃ c > 0, (fun n ↦ f k n) ≫ (fun (n : ℕ) ↦
    (n : ℝ) ^ (1 - 1 / ((k : ℝ) - 1)) / (log n) ^ c) := by
…
```
Mentions: c, Erdos920.f