Motivation comes from Sezeremide Theorem and Erdos discrapency
Let $G$ be the family of binary (or sign) sequences $s = (-1,1,1,..)$ satisfying the following property: $s$ has positive density and: For every $s \in G$ and every $k > 2\in N$, the subsequence $(s_{k}, s_{2k}, s_{3k},...)\in G $ as well, and is not equal to $s$.
Define a family $F$ as interesting if for all $s \in F $, there exist a sequence $s*$ that only disagrees with $s$ on one position and is not in $F$.
Then $G$ is not interesting