The board / group theory / #1098
Erdős Problem #1098
PROVED LEANdatabase updated 2026-05-02
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
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, \(gh \neq hg\).
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.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1098.Prizenone
Topicsgroup theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1098
Attempts
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