1098.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1098

References:

erdosproblems.com/1098
[Ne76] Neumann, B. H., A problem of Paul Erdős on groups. J. Austral. Math. Soc. Ser. A (1976), 467-472.

Imports (1)

FormalConjecturesUtil

def Erdos1098.nonCommutingGraph The non-commuting graph line 35 source
Source documentation
The non-commuting graph $\Gamma = \Gamma(G)$ of a group G, with vertices the elements of G and an edge between g and h if and only if g and h do not commute, $gh \neq hg$.

standard axioms: Quot.sound, propext what this means
Statement uses: Group, SimpleGraph
Proof uses: SimpleGraph.mk, Ne, HMul.hMul
8 instances, coercions and auxiliary constants left out.
```
def nonCommutingGraph (G : Type*) [Group G] : SimpleGraph G where
…
```
theorem Erdos1098.nonCommutingGraph_adj line 40 source
standard axioms: Quot.sound, propext what this means
Statement uses: Group, Iff, SimpleGraph.Adj, Erdos1098.nonCommutingGraph, Ne, HMul.hMul
Proof uses: Iff.rfl
6 instances, coercions and auxiliary constants left out.
theorem Erdos1098.erdos_1098 Documented declaration: erdos_1098 sorry line 55 source
Source documentation
Let $G$ be a group and $\Gamma=\Gamma(G)$ be the non-commuting graph, with vertices the elements of $G$ and an edge between $g$ and $h$ if and only if $g$ and $h$ do not commute, $gh\neq hg$.
If $\Gamma$ contains no infinite complete subgraph, then is there a finite bound on the size of complete subgraphs of $\Gamma$?
This was solved by Neumann [Ne76], who proved that $\Gamma$ contains no infinite complete subgraph if and only if the centre of the group has finite index, and noted that if the centre has index $n$ then $\Gamma$ contains no complete subgraph on $>n$ vertices.

depends on sorry what this means
Statement uses: Iff, True, Group, Set, SimpleGraph.IsClique, Erdos1098.nonCommutingGraph, Set.Finite, Exists, Nat, Finset, LE.le, Finset.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1098 : answer(True) ↔
    ∀ (G : Type*) [Group G],
      (∀ s : Set G, (nonCommutingGraph G).IsClique s → s.Finite) →
        ∃ n : ℕ, ∀ s : Finset G, (nonCommutingGraph G).IsClique (s : Set G) → s.card ≤ n := by
…
```
Mentions: Erdos1098.nonCommutingGraph
theorem Erdos1098.erdos_1098.variants.center_finiteIndex Neumann [Ne76] proved that $\Gamma$ contains no infinite complete subgraph if and only if the centre of the group has finite index sorry line 66 source
Source documentation
Neumann [Ne76] proved that $\Gamma$ contains no infinite complete subgraph if and only if the centre of the group has finite index.

depends on sorry what this means
Statement uses: Group, Iff, Set, SimpleGraph.IsClique, Erdos1098.nonCommutingGraph, Set.Finite, Subgroup.FiniteIndex, Subgroup.center
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1098.variants.center_finiteIndex (G : Type*) [Group G] :
    (∀ s : Set G, (nonCommutingGraph G).IsClique s → s.Finite) ↔
      (Subgroup.center G).FiniteIndex := by
…
```
Mentions: Erdos1098.nonCommutingGraph
theorem Erdos1098.erdos_1098.variants.upper_bound Neumann [Ne76] noted that if the centre has index $n$ then $\Gamma$ contains no complete subgraph on $>n$ vertices sorry line 76 source
Source documentation
Neumann [Ne76] noted that if the centre has index $n$ then $\Gamma$ contains no complete subgraph on $>n$ vertices.

depends on sorry what this means
Statement uses: Group, Subgroup.FiniteIndex, Subgroup.center, SimpleGraph.CliqueFree, Erdos1098.nonCommutingGraph, HAdd.hAdd, Nat, Subgroup.index, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1098.variants.upper_bound (G : Type*) [Group G]
    [(Subgroup.center G).FiniteIndex] :
    (nonCommutingGraph G).CliqueFree ((Subgroup.center G).index + 1) := by
…
```
Mentions: Erdos1098.nonCommutingGraph