628.lean`

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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 628

References:

erdosproblems.com/628
[BKPS09] Balogh, József and Kostochka, Alexandr V. and Prince, Noah and Stiebitz, Michael, The Erdős-Lovász Tihany conjecture for quasi-line graphs. Discrete Math. (2009), 3985-3991.
[BrJu69] Brown, W. G. and Jung, H. A., On odd circuits in chromatic graphs. Acta Math. Acad. Sci. Hungar. (1969), 129-134.
[Er68b] Erdős, P., Problem 2. Theory of Graphs (1968), 361.
[So22] Song, Zi-Xia, A survey on the Erdős-Lovász Tihany conjecture. Adv. Math. (China) (2022), 259--274.

Imports (1)

FormalConjecturesUtil

theorem Erdos628.erdos_628 Let $G$ be a graph with chromatic number $k$ containing no $K_k$ sorry line 46 source
Source documentation
Let $G$ be a graph with chromatic number $k$ containing no $K_k$. If $a,b\geq 2$ and $a+b=k+1$ then must there exist two disjoint subgraphs of $G$ with chromatic numbers $\geq a$ and $\geq b$ respectively?

depends on sorry what this means
Statement uses: Fintype, SimpleGraph, Nat, Eq, ENat, SimpleGraph.chromaticNumber, SimpleGraph.CliqueFree, GE.ge, OfNat.ofNat, HAdd.hAdd, Exists, Set, And, SimpleGraph.induce, Compl.compl
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_628 (V : Type*) [Fintype V] (G : SimpleGraph V) (k : ℕ)
    (hG_chrom : G.chromaticNumber = (k : ℕ∞))
    (hG_clique : G.CliqueFree k)
    (a b : ℕ) (ha : a ≥ 2) (hb : b ≥ 2) (hab : a + b = k + 1) :
    ∃ (s : Set V),
      (G.induce s).chromaticNumber ≥ (a : ℕ∞) ∧
      (G.induce sᶜ).chromaticNumber ≥ (b : ℕ∞) := by
…
```
theorem Erdos628.erdos_628.variants.k_5_a_3_b_3 Erdős [Er68b] originally asked about $a=b=3$ which was proved by Brown and Jung [BrJu69] (who in fact prove that $G$ must contain two vertex disjoint odd cycles) sorry line 61 source
Source documentation
Erdős [Er68b] originally asked about $a=b=3$ which was proved by Brown and Jung [BrJu69] (who in fact prove that $G$ must contain two vertex disjoint odd cycles)..

depends on sorry what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Eq, ENat, SimpleGraph.chromaticNumber, OfNat.ofNat, Nat, SimpleGraph.CliqueFree, Exists, Set, And, GE.ge, SimpleGraph.induce, Compl.compl
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_628.variants.k_5_a_3_b_3 (V : Type*) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG_chrom : G.chromaticNumber = (5 : ℕ∞))
    (hG_clique : G.CliqueFree 5) :
    ∃ (s : Set V),
      (G.induce s).chromaticNumber ≥ (3 : ℕ∞) ∧
      (G.induce sᶜ).chromaticNumber ≥ (3 : ℕ∞) := by
…
```
theorem Erdos628.erdos_628.variants.quasi_line Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for quasi-line graphs sorry line 73 source
Source documentation
Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for quasi-line graphs.

depends on sorry what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.IsQuasiLineGraph, Nat, Eq, ENat, SimpleGraph.chromaticNumber, SimpleGraph.CliqueFree, GE.ge, OfNat.ofNat, HAdd.hAdd, Exists, Set, And, SimpleGraph.induce, Compl.compl
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_628.variants.quasi_line (V : Type*) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG_quasi : IsQuasiLineGraph G)
    (k : ℕ)
    (hG_chrom : G.chromaticNumber = (k : ℕ∞))
    (hG_clique : G.CliqueFree k)
    (a b : ℕ) (ha : a ≥ 2) (hb : b ≥ 2) (hab : a + b = k + 1) :
    ∃ (s : Set V),
      (G.induce s).chromaticNumber ≥ (a : ℕ∞) ∧
      (G.induce sᶜ).chromaticNumber ≥ (b : ℕ∞) := by
…
```
theorem Erdos628.erdos_628.variants.independence_number_2 Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for graphs with independence number 2 sorry line 88 source
Source documentation
Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for graphs with independence number 2.

depends on sorry what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Eq, Nat, SimpleGraph.indepNum, OfNat.ofNat, ENat, SimpleGraph.chromaticNumber, SimpleGraph.CliqueFree, GE.ge, HAdd.hAdd, Exists, Set, And, SimpleGraph.induce, Compl.compl
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_628.variants.independence_number_2 (V : Type*) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG_indep : G.indepNum = 2)
    (k : ℕ)
    (hG_chrom : G.chromaticNumber = (k : ℕ∞))
    (hG_clique : G.CliqueFree k)
    (a b : ℕ) (ha : a ≥ 2) (hb : b ≥ 2) (hab : a + b = k + 1) :
    ∃ (s : Set V),
      (G.induce s).chromaticNumber ≥ (a : ℕ∞) ∧
      (G.induce sᶜ).chromaticNumber ≥ (b : ℕ∞) := by
…
```