Multiplicative representations of integers and Ramsey's theorem

Let ${\mathcal B} = (B_1,\ldots, B_h)$ be an $h$-tuple of sets of positive integers. Let $g_{\mathcal B}(n)$ count the number of representations of $n$ in the form $n = b_1\cdots b_h$, where $b_i \in B_i$ for all $i \in \{1,\ldots, h\}$. It is proved that $\liminf_{n\rightarrow \infty} g_{\mathcal B}(n) \geq 2$ implies $\limsup_{n\rightarrow \infty} g_{\mathcal B}(n) = \infty$.

Mel Nathanson