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Set in 1890, a London business man named Jabez Wilson, a man with red hair, comes to consult Detective Sherlock Holmes and his physician/side-kick Watson. He tells them that his young assistant, Vincent Spaulding, some weeks ago had shown him, and urged him to respond, to a newspaper want-ad offering work to only red-headed male applicants. The next morning, Wilson had waited in a long line of fellow red-headed men, was interviewed and was the only applicant hired, because none of the other applicants had hair to match Wilson's red locks.
Wilson, whose business mainly operates in evenings, was well-paid, receiving four pounds a week for several weeks; the work was obviously useless clerical work in a bare office. Finally one morning, a sign on the locked office door inexplicably announced that "THE RED-HEADED LEAGUE IS DISSOLVED."
Wilson then went to the landlord, who said that he'd never heard of Duncan Ross, the person who managed the league office. The landlord did remember the tenant with scarlet hair and gives him a card which directs Wilson to an artificial knee company. Wilson ends the story with how frustrated he is losing the four-pounds-a-week.
Holmes and Watson laugh a little over the ridiculous situation, but Holmes assures him that by Monday they would have the case solved. After Holmes' client, Wilson, leaves (having given the detective a description of Spaulding), Holmes decides to go and see Spaulding, whom Holmes notices has dirty trouser knees. Holmes then taps on the pavement in front of the pawnbroker's shop. With the case solved, he calls Police Inspector Jones and Mr. Merryweather, a director of the bank located next door.
The four secrete themselves in the bank vault and confront the thieves, John Clay, who has a long history of criminal activity already, and his helper Archie. Under the alias of Spaulding and Ross, they had contrived the "Red-Headed League" rigmarole to keep Wilson out of his shop while they dug in the basement, in order to break into the bank vault next door.
Plot summary from Wikipedia. Wikisource has the full story.