Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations

Abstract

Letting $\Omega(n)$ denote the number of prime factors of $n$ counted with multiplicity, Rivat, Sárközy and Stewart (1999) proved a result regarding maximal cardinalities of sets ${\cal A},{\cal B}\subset\{1,\ldots,N\}$ so that for every $a\in{\cal A}$ and $b\in{\cal B}$, $\Omega(a+b)$ is even. This paper extends their work in several directions. The role of $\lambda(n)=(-1)^{\Omega(n)}$ is generalized to all non-constant completely multiplicative functions $f:\mathbb{N}\rightarrow \{-1,1\}$. Rather than just $\Omega$ being even on ${\cal A}+{\cal B}$, we extend the result to all possible parities of $\Omega$ on ${\cal A}$, ${\cal B}$, and ${\cal A}+{\cal B}$. Furthermore, we prove that many such pairs $({\cal A},{\cal B})$ exist. Results from Ramsey theory and extremal graph theory are used.