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, $gh\ne hg$.If $\Gamma$ contains no infinite complete subgraph, then is there a finite bound on the size of complete subgraphs of $\Gamma$?