Novel structures in Stanley sequences

Given a set of integers with no 3-term arithmetic progression, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. This paper offers two main contributions to the theory of Stanley sequences. First, we describe all known Stanley sequences with closed-form expressions as solutions to constraints in modular arithmetic, defining the modular and pseudomodular Stanley sequences. Second, we introduce the basic Stanley sequences, whose elements arise as the sums of finite subsets of a basis   sequence. Applications of our results include the construction of Stanley sequences with arbitrarily large gaps between terms, answering a weak version of a problem by Erdős et al. Finally, we generalize several results about Stanley sequences to pp-free sequences, where pp is any odd prime.