On generalized Stanley sequences

Let N denote the set of all nonnegative integers. Let k ≥ 3 be an integer and A0 = {a1, . . . , at} (a1 < . . . < at) be a nonnegative set which does not contain an arithmetic progression of length k. We denote A = {a1, a2, . . . } defined by the following greedy algorithm: if l ≥ t and a1, . . . , al have already been defined, then al+1 is the smallest integer a > al such that {a1, . . . , al}∪{a} also does not contain a k-term arithmetic progression. This sequence A is called the Stanley sequence of order k generated by A0. In this paper, we prove some results about various generalizations of the Stanley sequence. 2010 Mathematics Subject Classification: Primary 11B75.