Erdős problem 1098 · problems.science

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, $gh\neq hg$ .

Sources

Browse retained paths and inspect the exact material available for this Problem.

4 retained statements2415f78e850a

Open selected source

FormalConjectures/ErdosProblems/

1098.lean

Retained formal statement1 of 4

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.

FormalConjectures/ErdosProblems/1098.leanErdos1098.erdos_10984 linesExact file

1True ↔2  ∀ (G : Type u_1) [inst : Group G],3    (∀ (s : Set G), (Erdos1098.nonCommutingGraph G).IsClique s → s.Finite) →4      ∃ n, ∀ (s : Finset G), (Erdos1098.nonCommutingGraph G).IsClique ↑s → s.card ≤ n

SolvedProof has a holelean4external proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.

Sources

Search problems.science