Extremal subsets of {1,...,n} avoiding solutions to linear equations in three variables

TL;DR

This paper refines previous results to classify extremal subsets of {1,...,n} that avoid solutions to certain non-invariant, homogeneous linear equations in three variables, enhancing understanding of their structure.

Contribution

It provides new examples and precise classifications of extremal subsets avoiding solutions to a broad class of linear equations in three variables.

Findings

Identification of extremal subsets avoiding specific linear equations

Examples and classifications of such subsets

Enhanced understanding of their structure and properties

Abstract

We refine previous results to provide examples, and in some cases precise classifications, of extremal subsets of {1,...,n} containing no solutions to a wide class of non-invariant, homogeneous linear equations in three variables, i.e.: equations of the form ax+by=cz with a+b \neq c.