The hex code of the day was #ff2400. Holmes and Watson, two prestigious men of their day, painted the walls of their study #ff2400. Together. They did a lot of things together, like solving murders without any definitive evidence. They also lived together.
They wanted to paint their study in a shade of red that sounded more pretentious than it looked. Scarlet. They now had a study in scarlet. It was beautiful.
Their study was almost as beautiful as Irene Adler, a woman they had met at the paint store downtown while looking for the perfect paint for the study they were going to paint scarlet. Or did they? Maybe they would meet her a full two years later. Doyle wasn’t very good with timelines or continuity. Most writers who don’t actually care about Sherlock Holmes aren’t good with continuity unless they pay attention. It’s hard to pay attention to characters you actively dislike writing.
With Sherlock Holmes’s job as a detective and John Watson’s job as a surgeon they didn’t really financially have cause to share a flat, but did anyway. Such was the life of Sherlock Holmes and the good doctor.
A woman died. Everyone thought it was suicide. Holmes and Watson didn’t think it was suicide because Holmes had super senses. He also had a good and healthy mistrust of the police. That is respectable, yet probably not a good idea when you’re inside a room full of them.
Holmes solved the murder with the help of Doctor Watson. Sadly, since all of the evidence was completely circumstantial, there were no witnesses, and most of the stuff Holmes said out loud sounded like utter nonsense, the killer got away.
That is how justice works. People who spew nonsense are treated as such, and detectives need to be able to actually prove things.