Let $F = \{f_1, ..., f_n\}$ be a set of distinct monic polynomials (and thus also $f_i^2$ are distinct). Let $F' = \{f_i f_j: i \neq j\}$ be the set of pairwise products of all distinct elements from $F$. We also require that $f_i^2 \neq f_j f_k$ for $j \neq k$.
Question. What is known about the upper and lower bounds on the cardinality of $F'$?
For example, if $F$ is a set of monomials, $f_i (X) = X^{t_i}$, then $F' = X^{t_i + t_j}$. The requirement above translates to the requirement that $(t_1, ..., t_n)$ does not have arithmetic progressions of length $3$. Therefore, one can take $(t_1, ..., t_n)$ to be equal to the progression-free set of Behrend or Elkind to minimize $|F'|$, and use the bound of Sanders to give a lower bound on $|F'|$.
But what happens in the general case?
The problem seems to have some relation to the results of Frankl-Wilson and Grolmusz on restricted intersections, but in our case $f_i^2$ are all distinct, while in their case (this is somewhat imprecise) $f_i^2 \equiv 0 \mod{p}$.