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run after someone
to chase someone of the opposite sex hoping for a date or some attention. Is John still running after Ann? No, Ann is running after John.
run after somebody
to try to get the attention or love of someone He's about to make a fool of himself, running after a married woman.
Etymology: based on the literal meaning of run after someone or something (to chase someone or something)
Also, chase after.
1. Follow, pursue with haste, as in Our dog loves to run after the mail truck, or The children were chasing after the geese in the park. [c. 1300]
2. Seek the company or attention of, especially aggressively. For example, He's run after her for a year, but she just ignores him. [Early 1500s]
1. To chase or pursue someone or something: The fox ran after the rabbit.
2. To follow something, always lagging behind: Stock prices have been running after bond prices for a while.
3. To seek the company or attention of someone for purposes of romance: She finally became tired of running after him. I can't believe the way he runs after her.