Independent sets in almost-regular graphs and the Cameron-Erdos problem for non-invariant linear equations

TL;DR

This paper generalizes the Cameron-Erdos conjecture to non-invariant linear equations over integers and proves a weak form for certain classes of these equations using graph theory methods.

Contribution

It extends the conjecture to a broader class of linear equations and applies graph theory techniques to establish partial results.

Findings

Proved a weak form of the generalized conjecture for specific equations.

Identified structural properties of maximum-size solution-avoiding sets.

Applied graph theory methods to problems in additive number theory.

Abstract

We propose a generalisation of the Cameron-Erdos conjecture for sum-free sets to arbitrary non-translation invariant linear equations over Z in three or more variables and, using well-known methods from graph theory, prove a weak form of the conjecture for a class of equations where the structure of the maximum-size sets avoiding solutions to the equation has been previously obtained.