The candidates names inside the cave, survivors have the numbers

Inside the cave, Locke tells Sawyer and all over the others are on the island because Jacob wrote his name on the cave walls/ceiling. He says that Jacob had a thing for numbers and each Locke, Reyes, Ford, Jarrah, Shephard and Kwon each correspond with on of the numbers. He says that he doesn't know if Kwon means Sun or Jin.

Locke tells Sawyer that he's sure that Sawyer met Jacob at some point when he was miserable or vulnerable. We saw that this happened in "The Incident" in season 5. Jacob met Sawyer at the funeral for his mother and father (see those screens here). Locke continues saying that Jacob manipulated Sawyer, using him as a puppet and pushing him towards the island. He says that Jacob did all of this because Sawyer and the others on the wall are/were candidates to take over Jacob's role as protector of the island.

Locke tells Saywer he has three choices: 1) He can do nothing and see how everything plays out and his name might get crossed out. 2) He can take the job and become the new Jacob. Locke says that it would be a waste of time because it's just and island and doesn't need Jacob, Sawyer or any of the others to protect it from anything. 3) He and Locke can get the hell off of the island together and never look back.

Sawyer chooses option three, joining team Locke/Smokey, but how much of what Locke told Sawyer was the truth? How much was his opinion, how much was lies and how much was the truth? Clearly, he disagreed with Jacob's views, but that doesn't mean that either one of them were necessarily right. We'll have to wait and see how this all plays out.

Kate's name is conspicuously missing from the wall or at least it is not visible in the scene, so why is she on the island?

Check out a list of decipherable names on the walls below. Click "Read more" below the list to see a whopping 46 screens.

4 - Locke (John)
8 - Reyes (Hugo "Hurley")
10 - Mattingly (name on one of the 1950s military uniforms)
15 - Ford (James "Sawyer")
16 - Jarrah (Sayid)
23 - Shephard (Jack)
31 - Rutherford (Shannon?)
42 - Kwon (Jin or Sun?)
64 - Goldstein
71 - Straume (Miles - who is crossed out for some reason)
90 - Troupe (Gary - the guy sucked into 815's engine)
117 - Linus (Ben or Roger?)
119 - Almieda
195 - Pace (Charlie)
222 - O'Toole
233 - Jones (name on one of the 1950s military uniforms)
291 - Domingo
313 - Littleton (Claire or Aaron?)
317 - Cunningham (name on one of the 1950s military uniforms)
346 - Grant
? - Chang (Pierre)
? - Goodspeed (Horace, Olivia or Ethan?)
? - Burke (Juliet)
? - Lacombe (one of the French team)
? - Lewis (Charlotte)
? - Faraday (Daniel)
? - Bargas
? - Aguila
? - Costa