Final answer:
The reticular formation is the central core structure that plays a key role in sleep and dreaming, especially during REM sleep.
Step-by-step explanation:
The structure in the central core that is involved in sleep and dreaming is the reticular formation. The thalamus helps regulate consciousness, arousal, and sleep states, but the pons, part of the brainstem, is primarily involved in regulating rapid eye movement (REM) sleep, which is closely associated with dreaming. While the medulla is also part of the brainstem, it is more focused on autonomic functions rather than sleep and dreaming specifically.