Step 1: Note what makes this interaction dangerous.
Theophylline has a narrow safety margin, so a small rise in blood level can cause toxicity. The question wants the drug that raises this level when a chest infection needs treatment.
Step 2: Think about enzyme inhibition.
The liver enzyme CYP1A2, with some help from CYP3A4, clears theophylline from the body. A drug that blocks this enzyme lets theophylline pile up.
Step 3: Match each option to its enzyme effect.
Erythromycin, a macrolide, is a well known inhibitor of CYP3A4 and partly CYP1A2. Ampicillin works on bacterial cell wall building and has no effect on liver enzymes. Cotrimoxazole targets bacterial folate synthesis and also spares CYP1A2. Sparfloxacin does not carry the strong CYP1A2 inhibition seen with ciprofloxacin or enoxacin.
Step 4: Connect the mechanism to the clinical picture.
Because erythromycin slows theophylline breakdown, giving it to this patient lets theophylline build up toward toxic levels, causing symptoms like vomiting, palpitations, and tremor.
Step 5: State the answer.
$Erythromycin$ is the antibiotic that can precipitate theophylline toxicity here.
\[ \boxed{\text{Erythromycin}} \]