On product Schur triples in the integers

TL;DR

This work studies product Schur triples $(a,b,c)$ with $ab=c$ in deterministic, random, and randomly perturbed environments. It connects to classical Schur theory through $S(k)$ and the double-sum number $S'(k)$ to derive deterministic bounds on the size of subsets that force monochromatic product triples, and to establish a $n^{1/3-\varepsilon}$ lower bound for the minimum number of such triples in 2-colourings. In the random model, it identifies the threshold for the appearance of a product Schur triple in $[2,n]_p$ as $(n\log n)^{-1/3}$, using first-moment and a two-copy construction for the upper bound. In the randomly perturbed model, it develops an $\\alpha$-perturbed framework and proves a Ford-divisor-based threshold interpolation between $n^{-1/2+o(1)}$ and $(n\log n)^{-1/3}$, governed by functions $f(\alpha)$ and $\beta(\alpha)$. Overall, the paper extends Schur-type phenomena to nonlinear multiplicative equations and highlights how density, randomness, and perturbations shape the emergence of monochromatic product structures with potential for further open questions.

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.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• proof : Proof of Theorem \ref{['thm:det_prod']}.
• Lemma 2.1
• Lemma 2.2
• proof : Proof of Lemma \ref{['thm:supersat']}
• proof : Proof of Theorem \ref{['thm:det_prod_mult']}