Monochromatic products in random integer sets
Roger Lidón, Darío Martínez, Patrick Morris, Miquel Ortega
TL;DR
This work initiates the study of Ramsey properties for nonlinear equations in random integer sets by focusing on monochromatic products. It establishes a 0-statement for two colours below $p rightarrow n^{-1/9-o(1)}$ via a greedy colouring that would fail only if forbidden configurations appear, and a 1-statement above $p ightarrow n^{-1/11}$ by embedding a fixed product-Schur pattern in [n]_p. For multiple colours, thresholds are bounded in terms of Schur-type numbers $S(r)$ and $S'(r)$, with precise asymptotics for small r and computational refinements suggesting improved constants. The results reveal a fundamentally different threshold behavior for nonlinear product configurations compared to linear Rado-type equations and open the door to broader nonlinear Ramsey considerations in random discrete structures.
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.