Multiplicative representations of integers and Ramsey's theorem

Abstract

Let $\mathcal{B}=(B_1,\dots ,B_h)$ 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 $i\in \{1,\dots ,h\}$. It is proved that ${\mathrm{liminf}}_{n\to \infty }{g}_{\mathcal{B}}(n)\ge 2$ implies ${\mathrm{limsup}}_{n\to \infty }{g}_{\mathcal{B}}(n)=\infty$.

11 pages

Cited by in corpus (2)

• On non-minimal complements
• $\mathbf{Q}$-independence and the construction of $B_h$-sets of integers and lattice points