762.lean`

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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 762

References:

erdosproblems.com/762
[EGS90] Erdős, Paul and Gimbel, John and Straight, H. Joseph, Chromatic number versus cochromatic number in graphs with bounded clique number. European J. Combin. (1990), 235-240.
[St24b] R. Steiner, On the difference between the chromatic and cochromatic number. arXiv:2408.02400 (2024).

Imports (1)

FormalConjecturesUtil

theorem Erdos762.erdos_762 Documented declaration: erdos_762 sorry line 46 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 empty graph.
Is it true that if $G$ has no $K_5$ and $\zeta(G)\geq 4$ then $\chi(G) \leq \zeta(G)+2$?
This has been disproved by Steiner [St24b], who constructed a graph $G$ with $\omega(G)=4$, $\zeta(G)=4$, and $\chi(G)=7$.

depends on sorry what this means
Statement uses: Iff, False, Fintype, SimpleGraph, SimpleGraph.CliqueFree, OfNat.ofNat, Nat, LE.le, ENat, SimpleGraph.cochromaticNumber, SimpleGraph.chromaticNumber, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_762 : answer(False) ↔
    ∀ (V : Type*) [Fintype V] (G : SimpleGraph V),
      G.CliqueFree 5 → 4 ≤ G.cochromaticNumber →
        G.chromaticNumber ≤ G.cochromaticNumber + 2 := by
…
```
theorem Erdos762.erdos_762.variants.bounded_clique_number Documented declaration: bounded_clique_number sorry line 57 source
Source documentation
A conjecture of Erdős, Gimbel, and Straight [EGS90], who proved that for every $n>2$ there exists some $f(n)$ such that if $G$ contains no clique on $n$ vertices then $\chi(G)\leq \zeta(G)+f(n)$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, Fintype, SimpleGraph, SimpleGraph.CliqueFree, LE.le, ENat, SimpleGraph.chromaticNumber, HAdd.hAdd, SimpleGraph.cochromaticNumber
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 erdos_762.variants.bounded_clique_number (n : ℕ) (hn : 2 < n) :
    ∃ f : ℕ, ∀ (V : Type*) [Fintype V] (G : SimpleGraph V),
      G.CliqueFree n → G.chromaticNumber ≤ G.cochromaticNumber + f := by
…
```
theorem Erdos762.erdos_762.variants.steiner This has been disproved by Steiner [St24b], who constructed a graph $G$ with $\omega(G)=4$, $\zeta(G)=4$, and $\chi(G)=7$ sorry line 67 source
Source documentation
This has been disproved by Steiner [St24b], who constructed a graph $G$ with $\omega(G)=4$, $\zeta(G)=4$, and $\chi(G)=7$.

depends on sorry what this means
Statement uses: Exists, Nat, SimpleGraph, Fin, And, Eq, SimpleGraph.cliqueNum, OfNat.ofNat, ENat, SimpleGraph.cochromaticNumber, SimpleGraph.chromaticNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_762.variants.steiner :
    ∃ (n : ℕ) (G : SimpleGraph (Fin n)),
      G.cliqueNum = 4 ∧ G.cochromaticNumber = 4 ∧ G.chromaticNumber = 7 := by
…
```