Let's examine each option to determine which one carries the same relationship as "Lazy : Listless."

1. Affection : Aversion - This pair does not have the same relationship as "Lazy : Listless." "Affection" and "aversion" are opposites, indicating a strong liking or love versus a strong disliking or avoidance. This is a contrast in emotions, not a similarity in characteristics like "Lazy : Listless."

2. Dusk : Dawn - This pair represents a relationship of opposites, much like "Affection : Aversion." "Dusk" is the period just before nightfall, and "dawn" is the period just before sunrise. They represent the beginning and end of the day, which are opposite phases of the day.

3. Foolish : Fatuous - This pair is similar to "Lazy : Listless" in that both sets of words describe related characteristics. "Foolish" and "fatuous" both mean lacking good sense or being silly. Therefore, this pair carries the same relationship as "Lazy : Listless."

4. Brainy : Barmy - This pair also does not have the same relationship as "Lazy : Listless." "Brainy" means intelligent or smart, whereas "barmy" means crazy or eccentric. These words describe contrasting characteristics rather than related ones.

So, the pair that carries the same relationship as "Lazy : Listless" is Option 3: Foolish : Fatuous because both pairs describe related characteristics or qualities.