Step 1: List the clinical features one by one.
Dizziness and vertigo point to the vestibular nuclei or their cerebellar connections.
Dysphagia points to the nucleus ambiguus, which supplies the pharynx and larynx.
Horner's syndrome (ptosis, miosis, anhidrosis) on the same side as the lesion points to interruption of the descending sympathetic fibers.
Loss of pain and temperature sensation on the opposite side of the body points to the spinothalamic tract, which has already crossed to the other side lower down in the spinal cord.
Step 2: Put the pattern together.
Ipsilateral cranial nerve and cerebellar signs together with contralateral pain and temperature loss in the body is the hallmark of a brainstem stroke affecting one side of the medulla, while the more medially placed corticospinal tract is largely spared. This combination is called lateral medullary syndrome, also known as Wallenberg syndrome.
Step 3: Identify the artery responsible.
The lateral medulla is supplied by the posterior inferior cerebellar artery (PICA), a branch of the vertebral artery. Occlusion of PICA is the classic cause of Wallenberg syndrome, producing exactly this mix, ipsilateral Horner's syndrome, ipsilateral vertigo and ataxia, dysphagia from nucleus ambiguus damage, and contralateral pain and temperature loss.
Step 4: Rule out the other arteries.
The anterior inferior cerebellar artery (AICA) supplies the lateral pons and causes a similar but higher syndrome, usually adding ipsilateral facial weakness and hearing loss, which are not the core description here.
The middle cerebral artery supplies the cerebral cortex and causes contralateral hemiparesis with cortical signs such as aphasia, not this brainstem pattern.
The superior cerebellar artery mainly causes ataxia and a different, less complete pattern, without the classic Horner's plus dysphagia combination.
Final Answer:
The clinical picture fits Wallenberg syndrome from a lateral medullary infarct, caused by the posterior inferior cerebellar artery.
\[ \boxed{\text{Posterior inferior cerebellar artery}} \]