147.lean`

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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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 147

References:

erdosproblems.com/147
[ErSi84] Erdős, P. and Simonovits, M., Cube-supersaturated graphs and related problems. Progress in graph theory (Waterloo, Ont., 1982) (1984), 203-218.
[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.
[Er97c] Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67.
[Ja23] Janzer, Oliver, Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles. Israel J. Math. (2023), 813--840.
[Ja23b] Janzer, Oliver, Disproof of a conjecture of Erdős and Simonovits on the Turán number of graphs with minimum degree 3. Int. Math. Res. Not. IMRN (2023), 8478--8494.

Imports (1)

FormalConjecturesUtil

theorem Erdos147.erdos_147 sorry line 67 source
depends on sorry what this means
Statement uses: Iff, False, Fintype, Nonempty, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.IsBipartite, LE.le, Nat, OfNat.ofNat, SimpleGraph.minDegree, Exists, Real, And, LT.lt, Asymptotics.IsBigO, Filter.atTop, HPow.hPow, HAdd.hAdd 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 Erdos147.erdos_147.variants.janzer_even Janzer [Ja23] disproved the conjecture for even sorry line 81 source
Source documentation
Janzer [Ja23] disproved the conjecture for even $r\geq 4$: for any even $r \geq 4$ and $\epsilon>0$ there exists a nonempty $r$-regular bipartite graph $H$ such that $\mathrm{ex}(n;H) \ll n^{2-\frac{2}{r}+\epsilon}$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Even, Real, LT.lt, Exists, SimpleGraph, Fin, And, SimpleGraph.IsBipartite, SimpleGraph.IsRegularOfDegree, Membership.mem, Set, SimpleGraph.neighborSet, Classical.propDecidable, SimpleGraph.Adj, Asymptotics.IsBigO, Filter.atTop, SimpleGraph.extremalNumber and 4 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_147.variants.janzer_even (r : ℕ) (hr : 4 ≤ r) (heven : Even r) (ε : ℝ)
    (hε : 0 < ε) :
    ∃ (q : ℕ) (H : SimpleGraph (Fin q)), 0 < q ∧ H.IsBipartite ∧ H.IsRegularOfDegree r ∧
      (fun n : ℕ => (extremalNumber n H : ℝ)) =O[atTop]
        fun n : ℕ => (n : ℝ) ^ (2 - 2 / (r : ℝ) + ε) := by
…
```
theorem Erdos147.erdos_147.variants.cubic Janzer [Ja23b] constructed, for any $\epsilon>0$, a nonempty $3$-regular bipartite graph $H$ such that $\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}$ sorry line 94 source
Source documentation
Janzer [Ja23b] constructed, for any $\epsilon>0$, a nonempty $3$-regular bipartite graph $H$ such that $\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Exists, Nat, SimpleGraph, Fin, And, SimpleGraph.IsBipartite, SimpleGraph.IsRegularOfDegree, Membership.mem, Set, SimpleGraph.neighborSet, Classical.propDecidable, SimpleGraph.Adj, Asymptotics.IsBigO, Filter.atTop, SimpleGraph.extremalNumber, HPow.hPow, HAdd.hAdd and 1 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 erdos_147.variants.cubic (ε : ℝ) (hε : 0 < ε) :
    ∃ (q : ℕ) (H : SimpleGraph (Fin q)), 0 < q ∧ H.IsBipartite ∧ H.IsRegularOfDegree 3 ∧
      (fun n : ℕ => (extremalNumber n H : ℝ)) =O[atTop]
        fun n : ℕ => (n : ℝ) ^ (4 / 3 + ε : ℝ) := by
…
```
theorem Erdos147.erdos_147.variants.janzer_conjecture Janzer [Ja23] conjectures that for any $r\geq 3$ and $\epsilon>0$ there exists a nonempty $r$-regular graph $H$ such that $\mathrm{ex}(n;H) \ll n^{2-\frac{2}{r}+\epsilon}$ sorry line 106 source
Source documentation
Janzer [Ja23] conjectures that for any $r\geq 3$ and $\epsilon>0$ there exists a nonempty $r$-regular graph $H$ such that $\mathrm{ex}(n;H) \ll n^{2-\frac{2}{r}+\epsilon}$.

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Real, LT.lt, Exists, SimpleGraph, Fin, And, SimpleGraph.IsRegularOfDegree, Membership.mem, Set, SimpleGraph.neighborSet, Classical.propDecidable, SimpleGraph.Adj, Asymptotics.IsBigO, Filter.atTop, SimpleGraph.extremalNumber and 4 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_147.variants.janzer_conjecture : answer(sorry) ↔ ∀ r : ℕ, 3 ≤ r → ∀ ε : ℝ, 0 < ε →
    ∃ (q : ℕ) (H : SimpleGraph (Fin q)), 0 < q ∧ H.IsRegularOfDegree r ∧
      (fun n : ℕ => (extremalNumber n H : ℝ)) =O[atTop]
        fun n : ℕ => (n : ℝ) ^ (2 - 2 / (r : ℝ) + ε) := by
…
```