Pursue someone romantically, as in We all thought Anne had set her cap for Joe, but we were wrong. In the 1700s this term, which may have alluded to donning one's best headgear, was applied to members of either sex, but by the early 1800s it generally described a woman chasing a man. It is probably obsolescent.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.