On solution-free sets of integers II

Given a linear equation L, a set A ⊆ [n] is L-free if A does not contain any ‘non-trivial’ solutions to L. We determine the precise size of the largest L-free subset of [n] for several general classes of linear equations L of the form px+qy = rz for fixed p, q, r ∈ N where p ≥ q ≥ r. Further, for all such linear equations L, we give an upper bound on the number of maximal L-free subsets of [n]. In the case when p = q ≥ 2 and r = 1 this bound is exact up to an error term in the exponent. We make use of container and removal lemmas of Green [12] to prove this result. Our results also extend to various linear equations with more than three variables.