On sets of integers with restrictions on their products

A product-injective labeling of a graph G is an injection χ:V(G)→Z such that χ(u)χ(v)≠χ(x)χ(y) for any distinct edges uv,xy∈E(G). Let P(G) be the smallest N≥1 such that there exists a product-injective labeling χ:V(G)→[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 drelative to n. Specifically, we show that there exist constants a,b>0 such that P(n,d)∼n if d≤n(logn)−a and P(n,d)∼nlogn if d≥n(logn)b.