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arXiv · 1101.0022

On the Growth of the Counting Function of Stanley Sequences

Richard A. Moy
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Abstract

Given a finite set of nonnegative integers A with no 3-term arithmetic progressions, the Stanley sequence generated by A, denoted S(A), is the infinite set created by beginning with A and then greedily including strictly larger integers which do not introduce a 3-term arithmetic progressions in S(A). Erdos et al. asked whether the counting function, S(A,x), of a Stanley sequence S(A) satisfies S(A,x)>x^{1/2-ε} for every ε>0 and x>x_0(ε,A). In this paper we answer this question in the affirmative; in fact, we prove the slightly stronger result that S(A,x)\geq (\sqrt{2}-ε)\sqrt{x} for x\geq x_0(ε,A).

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Richard A. Moy. 2012-02-03. On the Growth of the Counting Function of Stanley Sequences. https://doi.org/10.1016/j.disc.2010.12.019

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