Sets with no solutions to some symmetric linear equations

TL;DR

This paper introduces a new method for constructing large sets with no non-trivial solutions to certain symmetric linear equations, expanding known classes and providing probabilistic and explicit constructions.

Contribution

It develops a multi-dimensional construction technique for symmetric equations, allowing the creation of large solution-free sets for new classes of equations.

Findings

A new construction method for symmetric equations with no solutions.

Existence of near-optimal solution-free sets for randomly chosen symmetric equations.

Explicit constructions for symmetric equations in six variables.

Abstract

We expand the class of linear symmetric equations for which large sets with no non-trivial solutions are known. Our idea is based on first finding a small set with no solutions and then enlarging it to arbitrary size using a multi-dimensional construction, crucially assuming the equation in primitive. We start by presenting the technique on some new equations. Then we use it to show that a symmetric equation with randomly chosen coefficients has a near-optimal set with no non-trivial solutions. We also show a construction for a wide class of symmetric equation in 6 variables. In the final section we present a couple of remarks on non-symmetric equations.