Sárközy–Stewart conjecture on the greatest prime factor of products plus one

Let $N$ be an integer, and let $\mathcal A$ and $\mathcal B$ be subsets of $[1,N]$. Define

$$\mathcal C(\mathcal A,\mathcal B)=\{ab+1:a\in\mathcal A,\ b\in\mathcal B\}$$

and let $P^+(n)$ denote the greatest prime factor of $n$ for $n\geq 2$, with $P^+(1)=1$. Set

$$\varGamma^+(\mathcal A,\mathcal B,N)=\max_{c\in\mathcal C(\mathcal A,\mathcal B)}P^+(c).$$

Sárközy–Stewart conjecture. For every $\epsilon$ satisfying $0<\epsilon<1$, there exist $N(\epsilon)$ and $C(\epsilon)>0$ such that, for every integer $N\geq N(\epsilon)$ and every $\mathcal A,\mathcal B\subset[1,\dots,N]$ satisfying

$$|\mathcal A|,\ |\mathcal B|>\epsilon N,$$

we have

$$\varGamma^+(\mathcal A,\mathcal B,N)\geq C(\epsilon)N^2.$$

This conjecture predicts that dense subsets of $[1,N]$ always contain a product $ab+1$ with a prime factor of order $N^2$. The paper presents it as a stronger statement than the preceding lower bound, which reaches only order $N^{4/3}$ in the relevant range; its resolution is not supplied here.