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Longman Dictionary of Contemporary English
big name
noun
EXAMPLES FROM OTHER ENTRIES
▪ None of the soloists are big names yet, but there's a lot of talent there.
EXAMPLES FROM CORPUS
▪ He told me it was important for me to understand he was a very big name in his own town.
▪ The athlete-trainer had a successful business, a big name, a wife and a child in the Soviet Union.
Wiktionary
big name

n. 1 (context idiomatic English) A widely-known reputation, especially one that is favorable. 2 (context idiomatic English) A prominent individual, especially one who is favorably regarded.