760.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 760

References:

erdosproblems.com/760
[AKS97] Alon, Noga and Krivelevich, Michael and Sudakov, Benny, Subgraphs with a large cochromatic number. J. Graph Theory (1997), 295-297.

Imports (1)

FormalConjecturesUtil

theorem Erdos760.erdos_760 Documented declaration: erdos_760 sorry line 52 source
Source documentation
The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or independent set.
If $G$ is a graph with chromatic number $\chi(G)=m$ then must $G$ contain a subgraph $H$ with $$ \zeta(H) \gg \frac{m}{\log m}? $$
A problem of Erdős and Gimbel, who proved that there must exist a subgraph $H$ with $$ \zeta(H) \gg \left(\frac{m}{\log m}\right)^{1/2}. $$ The proposed bound would be best possible, as shown by taking $G$ to be a complete graph.
The answer is yes, proved by Alon, Krivelevich, and Sudakov.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Finite, SimpleGraph, Nat, Eq, ENat, SimpleGraph.chromaticNumber, SimpleGraph.Subgraph, LE.le, SimpleGraph.cochromaticNumber, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, HDiv.hDiv, HMul.hMul and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_760 : answer(True) ↔
    ∃ c > (0 : ℝ), ∀ (V : Type*) [Finite V] (G : SimpleGraph V) (m : ℕ),
      G.chromaticNumber = (m : ℕ∞) →
        ∃ H : G.Subgraph, ∃ k : ℕ, ((k : ℕ∞) ≤ H.coe.cochromaticNumber) ∧
          (c * (m : ℝ) / Real.log (m : ℝ) ≤ (k : ℝ)) := by
…
```
Mentions: c
theorem Erdos760.erdos_760.variants.erdos_gimbel A problem of Erdős and Gimbel, who proved that there must exist a subgraph $H$ with $$ \zeta(H) \gg \left(\frac{m}{\log m}\right)^{1/2}. $$ sorry line 66 source
Source documentation
A problem of Erdős and Gimbel, who proved that there must exist a subgraph $H$ with $$ \zeta(H) \gg \left(\frac{m}{\log m}\right)^{1/2}. $$

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Finite, SimpleGraph, Nat, Eq, ENat, SimpleGraph.chromaticNumber, SimpleGraph.Subgraph, LE.le, SimpleGraph.cochromaticNumber, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, HMul.hMul, Real.sqrt, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_760.variants.erdos_gimbel :
    ∃ c > (0 : ℝ), ∀ (V : Type*) [Finite V] (G : SimpleGraph V) (m : ℕ),
      G.chromaticNumber = (m : ℕ∞) →
        ∃ H : G.Subgraph, ∃ k : ℕ, ((k : ℕ∞) ≤ H.coe.cochromaticNumber) ∧
          (c * Real.sqrt ((m : ℝ) / Real.log (m : ℝ)) ≤ (k : ℝ)) := by
…
```
Mentions: c