Definition of come down with phrasal verb from the Oxford Advanced Learner's Dictionary

 

come down with

 phrasal verb
phrasal verb Ailments and diseases, Being ill
 
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Phrasal Verbs

come down with something

 
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[no passive] to get an illness that is not very serious I think I'm coming down with flu. See related entries: Ailments and diseases, Being ill
See the Oxford Advanced American Dictionary entry: come down with