The Coliseum is the one world you'll be coming back to time and time again. The Coliseum is the source of incredible amounts
of experience and some rare items, these are won through the various tournaments held there. For completing the Phil, Pegasus and Hercules cup you gain the Olympus Keychain.
Here's a nice list of when each Tournament becomes available.
Phil Cup: Right after you defeat the Guard Armor in Traverse Town
Pegasus Cup: After getting through Monstro
Hercules Cup: Right before Hollow Bastion
Hades Cup: After Hollow Bastion
Gold Match comes along after you complete the Hades Cup.
Platinum Match is after Hollow Bastion
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