  The clique number of a random graph in the Erdos-Renyi model G(n,p) yields a
random variable which is known to be asymptotically (as n tends to infinity)
almost surely within one of an explicit logarithmic (on n) function r(n,p). We
extend this fact by showing that random graphs have, asymptotically almost
surely, arbitrarily many pairwise disjoint complete subgraphs with as many
vertices as r(n,p). The result is motivated by and applied to the sequential
motion planning problem on random right angled Artin groups. Indeed, we give an
asymptotical description of all the higher topological complexities of
Eilenberg-MacLane spaces associated to random graph groups.
