Monochromatic products in random integer sets

TL;DR

This paper investigates the threshold probability at which random subsets of integers almost surely contain monochromatic solutions to the equation $ab=c$ under 2-colorings, revealing a non-linear threshold behavior.

Contribution

It establishes bounds for the threshold in random integer sets for monochromatic solutions to $ab=c$, highlighting differences from linear equation cases.

Findings

Threshold for 2-colorings lies between $n^{-1/9-o(1)}$ and $n^{-1/11}$.

Threshold behavior and proof methods differ from linear equation scenarios.

Non-linear equations exhibit distinct probabilistic thresholds compared to linear cases.

Abstract

A well-known consequence of Schur's theorem is that for $r\in \mathbb{N}$, if $n$ is sufficiently large, then any $r$-colouring of $[n]$ results in monochromatic $a,b,c\in [n]$ such that $ab=c$. In this paper we are interested in the threshold at which the binomial random set $[n]_p$ almost surely inherits this Ramsey-type property. In particular for $r=2$ colours, we show that this threshold lies between $n^{-1/9-o(1)}$ and $n^{-1/11}$. Whilst analogous questions for solutions to (sets of) linear equations are now well understood, our work suggests that both the behaviour of the thresholds and the proof methods needed to determine them differ substantially in the non-linear setting.