Enumerating solution-free sets in the integers

Abstract

Given a linear equation L, a set A⊆[n] is L-free if A does not contain any ‘non-trivial’ solutions to L. In this paper we consider the following three general questions:(i)What is the size of the largest L-free subset of [n]?(ii)How many L-free subsets of [n] are there?(iii)How many maximal L-free subsets of [n] are there? We completely resolve (i) in the case when L is the equation px+qy=z for fixed p,q∈N where p≥2. Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations L, thereby refining a special case of a result of Green [15]. We also give various bounds on the number of maximal L-free subsets of [n] for three-variable homogeneous linear equations L. For this, we make use of container and removal lemmas of Green [15].