A note on maximal progression-free sets

AbstractErdős et al [Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math. 200 (1999) 119–135.] asked whether there exists a maximal set of positive integers containing no three-term arithmetic progression and such that the difference of its adjacent elements approaches infinity. This note answers the question affirmatively by presenting such a set in which the difference of adjacent elements is strictly increasing. The construction generalizes to arithmetic progressions of any finite length

On generalized Stanley sequences

Let \mathbb{N} denote the set of all nonnegative integers. Let k \ge 3 be an integer and A_0 = {a_1,..., a_t} (a1 <...< at) be a nonnegative set which does not contain an arithmetic progression of length k. We denote A = {a_1, a_2,... } defined by the following greedy algorithm: if l ≥ t and a_1,..., a_l have already been defined, then a_{l+1} is the smallest integer a > a_l such that {a_1,..., a_l}\cup {a} also does not contain a k-term arithmetic progression. This sequence A is called the Stanley sequence of order k generated by A_0. In this paper, we prove some results about various generalizations of the Stanley sequence

On the classification of Stanley sequences

An integer sequence is said to be 3-free if no three elements form an arithmetic progression. Following the greedy algorithm, the Stanley sequence $S(a_0,a_1,\ldots,a_k)$ is defined to be the 3-free sequence $\{a_n\}$ having initial terms $a_0,a_1,\ldots,a_k$ and with each subsequent term $a_n>a_{n-1}$ chosen minimally such that the 3-free condition is not violated. Odlyzko and Stanley conjectured that Stanley sequences divide into two classes based on asymptotic growth patterns, with one class of highly structured sequences satisfying $a_n\approx \Theta(n^{\log_2 3})$ and another class of seemingly chaotic sequences obeying $a_n=\Theta(n^2/\log n)$. We propose a rigorous definition of regularity in Stanley sequences based on local structure rather than asymptotic behavior and show that our definition implies the corresponding asymptotic property proposed by Odlyzko and Stanley. We then construct many classes of regular Stanley sequences, which include as special cases all such sequences previously identified. We show how two regular sequences may be combined into another regular sequence, and how parts of a Stanley sequence may be translated while preserving regularity. Finally, we demonstrate that certain Stanley sequences possess proper subsets that are also Stanley sequences, a situation that appears previously to have been assumed impossible.Comment: 25 page