INMO 2023 Problem 1

Let $S$ be a finite set of positive integers. Assume that there are precisely 2023 ordered pairs $(x, y)$ in $S \times S$ so that the product $x y$ is a perfect square. Prove that one can find at least four distinct elements in $S$ so that none of their pairwise products is a perfect square.

Note: As an example, if $S=\{1,2,4\}$, there are exactly five such ordered pairs: $(1,1),(1,4)$, $(2,2),(4,1)$, and $(4,4)$.