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 p x + q y = 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].

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