Enumerating solution-free sets in the integers

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].