We answer a question of Sós by showing that, if a graph G of order n and density p has no complete minor larger than would be found in a random graph G(n, p), then G is quasi-random, provided either p > 0.45631 … or κ(G) [ges ] n(log log log n)/(log log n), where 0.45631 … is an explicit constant.

The results proved can also be used to fill the gaps in an argument of Thomason, describing the extremal graphs having no Kt minor for given t.