Multiplicative representations of integers and Ramsey's theorem

TL;DR

This paper investigates the behavior of multiplicative representations of integers across multiple sets, establishing a link between the liminf and limsup of the number of such representations.

Contribution

It proves that if the minimum number of representations is at least two infinitely often, then the maximum number of representations is unbounded.

Findings

If liminf of representations ≥ 2, then limsup is infinite.

Establishes a threshold behavior for multiplicative representations.

Provides a new connection between representation counts and Ramsey-type results.

Abstract

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 $\limsup_{n\rightarrow \infty} g_{\mathcal{B} }(n) = \infty$.