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

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 $⟨x,y⟩\in \mathfrak{X}\mathrm{.}$ Then either $G\in \mathfrak{X}$ or $\mathrm{\mid }G\mathrm{\mid }\le {\left(\frac{180}{53}\right)}^m(n-1)\mathrm{.}$