In the Alola games (Sun, Moon, and their respective remakes Ultra Sun and Ultra Moon), bringing a wild Pokémon to half health or lower can cause them to call for help, bringing in a second Pokémon to help them, usually of the same species, a member of their evolutionary line, or on occasion a completely different Pokémon. This can be very useful for completing the Pokédex, finding Pokémon with Hidden Abilities... or finding Shiny Pokémon. It's a careful dance to keep a call chain (or SOS chain as the community often refers to it) going long enough to find a Shiny (I've gone 200+ deep without luck and still hold a grudge against Bagon), but these instructions and tips should help you hold your own.
Calling for Help Won't Help You, but It'll Sure Help Me
Before you even get started, though, you need to come prepared - keeping the chain going may not take many Pokéballs, but you definitely need a lot of Leppa Berries. Why? Because in order to keep a chain going, you need to leave one Pokémon at less than half health so it can call for help, then knock out the Pokémon it summons so it can call for another, and even if you PP Max all of your party's moves so each one can attack over a hundred times... bad RNG can still lead to you running out of moves and failing the chain. Poké Pelago's Isle Aplenny lets you farm Berries risk-free (as opposed to older games where if you forget to pick your plants, they can die), so you can end up with hundreds of Leppa Berries very easily, leading to you having thousands of attacks with a single party member.
As for the specific Pokémon that you bring... there are several strategies out there to make chaining more efficient and/or easier, but all you really need is a Pokémon with False Swipe, a Pokémon strong enough to not need healing often and can KO the target Pokémon and any Pokémon it can call with one attack (some Pokémon call allies stronger than themselves; Eevee immediately comes to mind when calling Espeon / Umbreon but there are others), and a Pokémon with a catching-oriented moveset (False Swipe, Thunder Wave / Spore, Recover, Swords Dance for example; can be the same as the first Pokémon).
There's one last thing you'll need before you start chaining, and that's an Adrenaline Orb. This item is the key to chaining, since without it, wild Pokémon will only call for help once before stopping. You can buy them fairly cheaply and fairly early in the game, but they're pretty easy to forget about. Once you have your team, the load of Leppa Berries, and the Adrenaline Orb, you're ready to go!
To start the chain, simply enter a battle with the Pokémon you intend to chain, leading with your False Swipe user. Immediately use the Adrenaline Orb; the battle text will say that the Pokémon grew nervous. Use False Swipe until the Pokémon is at 1 HP, then switch to your attacker, knocking out the allies as the wild Pokémon calls them. However, you must keep track of how many times the Pokémon attacks; if it uses all of its PP, then it will Struggle and knock itself out, breaking the chain. I recommend KO'ing the original Pokémon when a Pokémon of the same species is on the field, otherwise the chain breaks around once every 40-50 turns for most Pokémon, though some Pokémon with less moves need to be switched more frequently. Repeat this process until a Shiny appears, knock out the non-Shiny Pokémon, and bring out your catcher to finally catch the Shiny.
By now, I'm sure you're wondering what the Shiny odds are for SOS chaining, especially considering how much work and attention needs to be put into it. Thankfully, the odds start shooting up almost immediately; only ten "links" (or Pokémon knocked out) already sees a notable increase in odds, and the cap is a fairly low 30-chain, as shown in the table.
