Applying the K\"ov\'ari-S\'os-Tur\'an theorem to a question in group theory

Abstract

Let $m\leq n$ 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 $\langle x, y \rangle\in \mathfrak X.$ Then either $G\in \mathfrak X$ or $|G|\leq \left(\frac{180}{53}\right)^m(n-1).$