to run to try to catch sb / sth; to try to have a romantic or sexual relationship with sb
Oxford71% of films
to run to try to catch sb / sth
You run after them, I'll cut them off.
Oxford14% of films
to try to have a romantic or sexual relationship with sb
You told me, you wanted to run after that girl.
Definitions from the Oxford Advanced Learner's Dictionary, 9th edition