Exact Lower Bounds for Monochromatic Schur Triples and Generalizations

TL;DR

This paper establishes exact and sharp lower bounds for monochromatic generalized Schur triples in two-colorings of sets, extending previous asymptotic results to arbitrary real parameters and providing explicit formulas for specific cases.

Contribution

It introduces a method to derive exact bounds for generalized Schur triples for any real parameter, advancing beyond prior asymptotic formulas and specific integer cases.

Findings

Exact lower bounds for generalized Schur triples derived

Explicit formulas provided for a=1,2,3,4 cases

Extension of results to arbitrary real a using symbolic computation

Abstract

We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots,n\}$, subject to a coloring with two different colors. Previously, only asymptotic formulas for such bounds were known, and only for $a\in\mathbb{N}$. Using symbolic computation techniques, these results are extended here to arbitrary $a\in\mathbb{R}$. Furthermore, we give exact formulas for the minimum number of monochromatic Schur triples for $a=1,2,3,4$, and briefly discuss the case $0<a<1$.