On sets of integers with restrictions on their products

A {\em product-injective labeling} of a graph $G$ is an injection $\chi: V(G) \to \mathbb{Z}$ such that $\chi(u)\chi(v) \not= \chi(x)\chi(y)$ for any distinct edges $uv, xy\in E(G)$. Let $P(G)$ be the smallest $N \geq 1$ such that there exists a product-injective labeling $\chi : V(G) \rightarrow [N]$. Let $P(n,d)$ be the maximum possible value of $P(G)$ over $n$-vertex graphs $G$ of maximum degree at most $d$. In this paper, we determine the asymptotic value of $P(n,d)$ for all but a small range of values of $d$ relative to $n$. Specifically, we show that there exist constants $a,b > 0$ such that $P(n,d) \sim n$ if $d \leq \sqrt{n}(\log n)^{-a}$ and $P(n,d) \sim n\log n$ if $d \geq \sqrt{n}(\log n)^{b}$.