Applying the Kövári-Sós-Turán theorem to a question in group theory

Abstract

Let $m\le n$ be positive integers and $\mathfrak{X}$ a class of groups which is closed for subgroups, quotient groups and extensions. Suppose that a finite group $G$ satisfies the condition that for every two subsets $M$ and $N$ of cardinalities $m$ and $n,$ respectively, there exist $x\in M$ and $y\in N$ such that $⟨x,y⟩\in \mathfrak{X}.$ Then either $G\in \mathfrak{X}$ or $|G|\le {\left(\frac{180}{53}\right)}^m(n-1).$