559.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

6 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 559

References:

erdosproblems.com/559
[Be83b] Beck, József, On size Ramsey number of paths, trees, and circuits. I. J. Graph Theory (1983), 115-129.
[FrPi87] Friedman, J. and Pippenger, N., Expanding graphs contain all small trees. Combinatorica (1987), 71-76.
[HKL95] Haxell, P. E. and Kohayakawa, Y. and Łuczak, T., The induced size-Ramsey number of cycles. Combin. Probab. Comput. (1995), 217-239.
[RoSz00] Rödl, Vojtěch and Szemerédi, Endre, On size Ramsey numbers of graphs with bounded degree. Combinatorica (2000), 257-262.
[Ti22b] Tikhomirov, K., On bounded degree graphs with large size-Ramsey numbers. arXiv:2210.05818 (2022).
[DrPe22] N. Draganić and K. Petrova, Size-Ramsey numbers of graphs with maximum degree three. arXiv:2207.05048 (2022).

Imports (1)

FormalConjecturesUtil

theorem Erdos559.erdos_559 sorry line 59 source
depends on sorry what this means
Statement uses: Iff, False, Nat, Exists, Real, Fintype, SimpleGraph, LE.le, SimpleGraph.maxDegree, Classical.propDecidable, SimpleGraph.Adj, SimpleGraph.sizeRamsey, HMul.hMul, Fintype.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos559.erdos_559.variants.degree_three sorry line 68 source
depends on sorry what this means
Statement uses: Not, Exists, Real, Fintype, SimpleGraph, LE.le, Nat, SimpleGraph.maxDegree, Classical.propDecidable, SimpleGraph.Adj, OfNat.ofNat, SimpleGraph.sizeRamsey, HMul.hMul, Fintype.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos559.erdos_559.variants.rodl_szemeredi Rödl and Szemerédi [RoSz00] constructed, for infinitely many sorry line 78 source
Source documentation
Rödl and Szemerédi [RoSz00] constructed, for infinitely many $n$, a graph on $n$ vertices with maximum degree $3$ such that $\hat{R}(G)\gg n(\log n)^{c}$ for some absolute constant $c>0$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Frequently, Nat, SimpleGraph, Fin, LE.le, SimpleGraph.maxDegree, Classical.propDecidable, SimpleGraph.Adj, HMul.hMul, HPow.hPow, Real.log, SimpleGraph.sizeRamsey, 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_559.variants.rodl_szemeredi :
    ∃ c > (0 : ℝ), ∃ C > 0, ∃ᶠ n : ℕ in atTop, ∃ G : SimpleGraph (Fin n),
      G.maxDegree ≤ 3 ∧ C * n * (log n) ^ c ≤ sizeRamsey G G := by
…
```
Mentions: c
theorem Erdos559.erdos_559.variants.tikhomirov Tikhomirov [Ti22b] improved this to $\hat{R}(G)\gg n\exp(c\sqrt{\log n})$ sorry line 86 source
Source documentation
Tikhomirov [Ti22b] improved this to $\hat{R}(G)\gg n\exp(c\sqrt{\log n})$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Frequently, Nat, SimpleGraph, Fin, LE.le, SimpleGraph.maxDegree, Classical.propDecidable, SimpleGraph.Adj, HMul.hMul, Real.exp, Real.sqrt, Real.log, SimpleGraph.sizeRamsey, Filter.atTop
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_559.variants.tikhomirov :
    ∃ c > 0, ∃ C > 0, ∃ᶠ n : ℕ in atTop, ∃ G : SimpleGraph (Fin n),
      G.maxDegree ≤ 3 ∧ C * n * exp (c * √(log n)) ≤ sizeRamsey G G := by
…
```
Mentions: c
theorem Erdos559.erdos_559.variants.trees Friedman and Pippenger [FrPi87] proved the linear bound when $G$ is a tree sorry line 94 source
Source documentation
Friedman and Pippenger [FrPi87] proved the linear bound when $G$ is a tree.

depends on sorry what this means
Statement uses: Nat, Exists, Real, Fintype, SimpleGraph, SimpleGraph.IsTree, LE.le, SimpleGraph.maxDegree, Classical.propDecidable, SimpleGraph.Adj, SimpleGraph.sizeRamsey, HMul.hMul, Fintype.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_559.variants.trees (d : ℕ) :
    ∃ C : ℝ, ∀ (V : Type) [Fintype V] (G : SimpleGraph V),
      G.IsTree → G.maxDegree ≤ d → (sizeRamsey G G : ℝ) ≤ C * Fintype.card V := by
…
```
theorem Erdos559.erdos_559.variants.draganic_petrova The best known upper bound for graphs of maximum degree $3$ is $\hat{R}(G)\leq n^{3/2+o(1)}$, due to Draganić and Petrova [DrPe22] sorry line 103 source
Source documentation
The best known upper bound for graphs of maximum degree $3$ is $\hat{R}(G)\leq n^{3/2+o(1)}$, due to Draganić and Petrova [DrPe22].

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, LE.le, SimpleGraph.maxDegree, Classical.propDecidable, SimpleGraph.Adj, SimpleGraph.sizeRamsey, HPow.hPow, HAdd.hAdd, HDiv.hDiv, Filter.atTop
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_559.variants.draganic_petrova (ε : ℝ) (hε : 0 < ε) :
    ∀ᶠ n : ℕ in atTop, ∀ G : SimpleGraph (Fin n),
      G.maxDegree ≤ 3 → (sizeRamsey G G : ℝ) ≤ (n : ℝ) ^ (3 / 2 + ε : ℝ) := by
…
```