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, $gh\neq hg$.

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.