The central conflict in Sir Arthur Conan Doyle's story "The Red-Headed League" is between John Clay and the detective Sherlock Holmes. Clay wants to get away with robbing a bank undiscovered and Holmes wants to solve the mystery of who robbed the bank, which would cause him to apprehend Clay.
Clay is a career criminal whose past crimes include robbery and murder. He has an aristocratic background and may have embarked on a life of crime more for its entertainment value than financial gain. In this story, Clay initially appears under the pseudonym Spaulding and works as a clerk for Jabez Wilson. He is notable for his intelligence as well as a philanthropic streak.
Holmes is a consulting detective who is a recurring character in many of Doyle's works. He uses the science of deduction, an intimate knowledge of London, and forensic science to solve crimes. Like Clay, his motive is mainly to entertain himself and avoid boredom.
The main conflict is a character vs. character external conflict: Holmes vs. Clay.
Conflicts are internal or external. An internal conflict is one a character has with him or her self.
In this story, the main conflict is an external one between Holmes and John Clay. Clay does not know what Holmes is chasing him, and in fact Holmes does not know that Clay is robbing a bank until Mr. Wilson comes to hire him.
I made inquiries as to this mysterious assistant and found that I had to deal with one of the coolest and most daring criminals in London. (p. 18)
The conflict between Holmes and Clay is an external conflict. In the end, each agrees that the other was a worthy opponent.