On Generalized Schur Numbers of the Equation x+ay=z

Amir Khamseh

doi:10.1155/2020/7069730

Abstract

Let a and r be positive integers. By definition, sar is the least positive integer such that, for any r-coloring of the interval 1,sar, there exists a monochromatic solution to x+ay=z. For a=1, the numbers sr=s1r are classical Schur numbers. In this paper, we study the numbers sar for a≥2.

Highlights

An area that has seen a remarkable burst of research activity during the past twenty years, is the study of the preservation of properties under set partitions
Given a particular set S that has a property P, is it true that whenever S is partitioned into finitely many subsets, one of the subsets must have property P? Ramsey theory is named after Frank Plumpton Ramsey and his theorem, which he proved in 1928
For more on Ramsey theory and its applications, we refer to the book of Graham et al, [6], and the surveys of Nešetřil [7] and Rosta [8]

Summary

Introduction

An area that has seen a remarkable burst of research activity during the past twenty years, is the study of the preservation of properties under set partitions. The result that is generally accepted to be the first Ramsey-type theorem is due to Schur [1] and it deals with colorings of the integers: if N is partitioned into a finite number of classes, at least one partition class contains a solution to the equation x + y z. Let S(N) be the minimum number of monochromatic Schur triples in any 2coloring of [N] {1, 2, . For any r ≥ 1, there exists n such that for every r-coloring of [1, n], there is a monochromatic solution to the linear equation 􏽐ni 1 ci xi 0, where ci ∈ Z − {0} for 1 ≤ i ≤ n, if and only if some nonempty subset of the ci ’s sums to 0

Exact Value of sa

A General Lower Bound

Concluding Remark