22.lean`

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 22

The central problem of Ramsey–Turán theory: writing $\mathrm{rt}(n; k, \ell)$ for the maximum number of edges of a $K_k$-free graph on $n$ vertices whose largest independent set has size less than $\ell$, is it true that for every $\epsilon > 0$ and all sufficiently large $n$ $$\mathrm{rt}(n; 4, \epsilon n) \geq n^2/8?$$

Conjectured by Bollobás and Erdős [BoEr76], who constructed such a graph with $(1/8 + o(1))n^2$ edges. Together with the matching upper bound $\mathrm{rt}(n; 4, \epsilon n) \leq (1/8 + o(1))n^2$ of Szemerédi [Sz72], this determines the Ramsey–Turán density of $K_4$ to be $1/8$. The conjecture as stated was proved by Fox, Loh, and Zhao [FLZ15].

References:

erdosproblems.com/22
[BoEr76] Bollobás, B. and Erdős, P., On a Ramsey-Turán type problem. J. Combin. Theory Ser. B 21 (1976), 166--168.
[Sz72] Szemerédi, E., On graphs containing no complete subgraph with 4 vertices (Hungarian). Mat. Lapok 23 (1972), 113--116.
[FLZ15] Fox, J., Loh, P.-S., and Zhao, Y., The critical window for the classical Ramsey-Turán problem. Combinatorica 35 (2015), 435--476.

Imports (1)

FormalConjecturesUtil

theorem Erdos22.erdos_22 sorry line 60 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Exists, SimpleGraph, Fin, And, SimpleGraph.CliqueFree, LE.le, SimpleGraph.indepNum, HMul.hMul, HDiv.hDiv, HPow.hPow, Finset.card, Sym2, SimpleGraph.edgeFinset and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos22.erdos_22.variants.szemeredi_upper The matching upper bound, due to Szemerédi [Sz72]: a $K_4$-free graph on $n$ vertices whose independence number is sublinear in $n$ has at most $(1/8 + o(1))n^2$ edges sorry line 75 source
Source documentation
The matching upper bound, due to Szemerédi [Sz72]: a $K_4$-free graph on $n$ vertices whose independence number is sublinear in $n$ has at most $(1/8 + o(1))n^2$ edges. That is, for every $\epsilon > 0$ there is a $\delta > 0$ such that for all sufficiently large $n$, every $K_4$-free graph $G$ on $n$ vertices with $\alpha(G) \leq \delta n$ has at most $(1/8 + \epsilon)n^2$ edges.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Exists, And, Filter.Eventually, Nat, SimpleGraph, Fin, SimpleGraph.CliqueFree, LE.le, SimpleGraph.indepNum, HMul.hMul, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, HAdd.hAdd, HDiv.hDiv and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_22.variants.szemeredi_upper (ε : ℝ) (hε : 0 < ε) :
    ∃ δ : ℝ, 0 < δ ∧ ∀ᶠ (n : ℕ) in atTop,
      ∀ G : SimpleGraph (Fin n), G.CliqueFree 4 →
        (G.indepNum : ℝ) ≤ δ * n →
        (G.edgeFinset.card : ℝ) ≤ (1 / 8 + ε) * n ^ 2 := by
…
```
theorem Erdos22.erdos_22.variants.bollobas_erdos_lower Documented declaration: bollobas_erdos_lower sorry line 90 source
Source documentation
The construction of Bollobás and Erdős [BoEr76]: for every $\epsilon > 0$ and $\delta > 0$, for all sufficiently large $n$ there is a $K_4$-free graph on $n$ vertices with independence number at most $\delta n$ and at least $(1/8 - \epsilon)n^2$ edges. Together with erdos_22.variants.szemeredi_upper this shows that the Ramsey–Turán density of $K_4$ is $1/8$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Exists, SimpleGraph, Fin, And, SimpleGraph.CliqueFree, LE.le, SimpleGraph.indepNum, HMul.hMul, HSub.hSub, HDiv.hDiv, HPow.hPow, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_22.variants.bollobas_erdos_lower (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) :
    ∀ᶠ (n : ℕ) in atTop,
      ∃ G : SimpleGraph (Fin n), G.CliqueFree 4 ∧
        (G.indepNum : ℝ) ≤ δ * n ∧ (1 / 8 - ε) * n ^ 2 ≤ G.edgeFinset.card := by
…
```
theorem Erdos22.erdos_22.variants.fox_loh_zhao The quantitative strengthening proved by Fox, Loh, and Zhao [FLZ15]: there is a constant sorry line 104 source
Source documentation
The quantitative strengthening proved by Fox, Loh, and Zhao [FLZ15]: there is a constant $C > 0$ such that for all sufficiently large $n$ there exists a $K_4$-free graph on $n$ vertices with at least $n^2/8$ edges whose largest independent set has size at most $$C \cdot \frac{(\log\log n)^{3/2}}{(\log n)^{1/2}} \cdot n.$$

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, SimpleGraph.CliqueFree, LE.le, SimpleGraph.indepNum, HMul.hMul, HDiv.hDiv, HPow.hPow, Real.log, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_22.variants.fox_loh_zhao :
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ (n : ℕ) in atTop,
      ∃ G : SimpleGraph (Fin n), G.CliqueFree 4 ∧
        (G.indepNum : ℝ) ≤
          C * (Real.log (Real.log n)) ^ (3 / 2 : ℝ) / (Real.log n) ^ (1 / 2 : ℝ) * n ∧
        (n : ℝ) ^ 2 / 8 ≤ G.edgeFinset.card := by
…
```
theorem Erdos22.erdos_22.variants.test_bot A sanity check for erdos_22.variants.szemeredi_upper: the empty graph is $K_4$-free and trivially satisfies the upper bound $(1/8 + \epsilon)n^2$ on the number of edges line 117 source
Source documentation
A sanity check for erdos_22.variants.szemeredi_upper: the empty graph is $K_4$-free and trivially satisfies the upper bound $(1/8 + \epsilon)n^2$ on the number of edges.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LT.lt, OfNat.ofNat, And, SimpleGraph.CliqueFree, Fin, Bot.bot, SimpleGraph, LE.le, Finset.card, Sym2, SimpleGraph.edgeFinset, HMul.hMul, HAdd.hAdd, HDiv.hDiv, HPow.hPow
Proof uses: And.intro, SimpleGraph.cliqueFree_bot, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Eq.mpr, EmptyCollection.emptyCollection, Finset, id, Eq, congrArg, SimpleGraph.edgeFinset_bot, congrFun', Nat.cast_zero, mul_nonneg, le_of_lt, add_pos', div_pos and 6 more
109 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_22.variants.test_bot (n : ℕ) (ε : ℝ) (hε : 0 < ε) :
    (⊥ : SimpleGraph (Fin n)).CliqueFree 4 ∧
      (((⊥ : SimpleGraph (Fin n)).edgeFinset.card : ℝ)) ≤ (1 / 8 + ε) * n ^ 2 := by
…
```