# 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.

## Key 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 $0<a<1$.

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Source: https://tomesphere.com/paper/1904.01925