On product Schur triples in the integers

Abstract

Schur's theorem states that in any $k$-colouring of the set of integers $[n]$ there is a monochromatic solution to $a+b=c$, provided $n$ is sufficiently large. Abbott and Wang studied the size of the largest subset of $[n]$ such that there is a $k$-colouring avoiding a monochromatic $a+b=c$. In other directions, the minimum number of $a+b=c$ in $k$-colourings of $[n]$ and the probability threshold in random subsets of $[n]$ for the property of having a monochromatic $a+b=c$ in any $k$-colouring were investigated. In this paper, we study natural generalisations of these streams to products $ab=c$, in a deterministic, random, and randomly perturbed environments.