Stanley sequences with odd character

Abstract

Given a set of integers containing no 3-term arithmetic progressions, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. Independent Stanley sequences are a "well-structured" class of Stanley sequences with two main parameters: the character $\lambda (A)$ and the repeat factor $\rho (A)$. Rolnick conjectured that for every $\lambda \in {\mathbb{N}}_0\backslash \{1,3,5,9,11,15\}$, there exists an independent Stanley sequence $S(A)$ such that $\lambda (A)=\lambda$. This paper demonstrates that $\lambda (A)\notin \{1,3,5,9,11,15\}$ for any independent Stanley sequence $S(A)$.