run after

run after
PURSUE, chase; make advances to, flirt with; informal make up to, come on to, be all over, vamp; dated set one's cap at.

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informal seek to acquire or attain; pursue persistently

businesses that have spent years running after the boomer market

seek the company of (someone) with the aim of developing a romantic or sexual relationship with them

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run after [phrasal verb]
1 run after (someone or something) : to run toward (someone or something) in an attempt to catch that person or thing

The dog ran after [=chased] the squirrel.

2 run after (someone) informal + disapproving : to try to start a romantic relationship with (someone)

He's always running after younger women.

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Main Entry:run

Useful english dictionary. 2012.