Of the Sherlock Holmes short stories, what is the minimum number necessary to outline Holmes' career?
"A Scandal in Bohemia" and "His Last Bow" are the beginning and the end. The latter appropriately refers back to the former.
"The Final Problem" and "The Empty House" are the middle: the "death" and return.
"The Greek Interpreter," set before the "death," introduces Mycroft Holmes. "The Bruce-Partington Plans," set after the return, explains Mycroft's role in the Government.
"His Last Bow" is a return from the retirement that is announced in "The Second Stain" and described in "The Lion's Mane."
Therefore, I think these eight are the minimum. "The Bruce-Partington Plans" and "The Second Stain" name certain foreign agents working in London. This links these stories directly to each other and indirectly to "His Last Bow" in which Holmes and Watson apprehend Von Bork who is as significant in international espionage as Moriarty was in national crime.
I would also include "The Red-headed League" and "The Naval Treaty." John Clay, who is behind the Red-headed League hoax, is high in London crime which means that Holmes must be well on the road to identifying Moriarty. At least two dramatizations have retroactively linked Moriarty to the Red-headed League. Holmes says that Clay is "the fourth smartest man in London." The three smartest must be Moriarty, Mycroft and Sherlock although Doyle was probably not thinking that when he wrote this story.
"The Naval Treaty," coming between "The Greek Interpreter" and "The Final Problem," is a third story about an important stolen document and thus fits in with "The Bruce-Partington Plans" and "The Second Stain."
If we include the novels, then we must include them all. Holmes and Watson meet in A Study in Scarlet. Watson marries in The Sign of Four, thus explaining why he is not living at Baker Street in many of the stories starting with "A Scandal in Bohemia." The Valley of Fear, set before "The Final Problem," features a criminal gang who, although they are brought to justice, engage Moriarty to exact their revenge. Finally, The Hound of the Baskervilles, perhaps the novel that has been most often filmed, cannot be left out of any list of Holmes stories.