Stanley sequences with odd character

TL;DR

This paper investigates the structure of independent Stanley sequences, proving that certain character values are impossible, thereby advancing understanding of their parameter constraints.

Contribution

It demonstrates that specific character values cannot occur for any independent Stanley sequence, refining the classification of these sequences.

Findings

Certain character values are impossible for independent Stanley sequences.

The paper refutes a conjecture about the existence of sequences with specific characters.

Provides new insights into the structure and limitations of Stanley sequences.

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 $\lambda(A) \not\in \{1, 3, 5, 9, 11, 15\}$ for any independent Stanley sequence $S(A)$.