On generalized Stanley sequences

Kiss Z., Sándor and Sándor, Csaba and Yang, Quan-Hui (2018) On generalized Stanley sequences. Acta Mathematica Hungarica, 154 (2). pp. 501-510. ISSN 0236-5294

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Abstract

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.

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