A problem of Paul Erdös on groups

Abstract

Let G be a group, and associate with G a graph 1 γ = Γ (G) as follows: the vertices of Γ are the elements of G, and two vertices g, h of Γ are joined by an undirected edge if, and only if, g and h do not commute as elements of G, that is [g, h] ≠ 1 [where [g, h] is the commutator g−1h−1gh, and 1 is the unit element of the groups that occur as well as the integer, according to context].We are interested in complete subgraphs of Γ, or equivalently in sets of elements of G no two of which commute.