All the samples that pertain to achievements, leaderboards, and game center authenticated are obsolete.
If you search the sample code in the documentation library on "GK" you get samples that are from 2009 to 2011:
GKLeaderboards, GKAchievements, GKAuthentication, GKRocket, GKTapper, GKTank.
When I load one of these in Xcode, I get deprecation warnings, which is not surprising of course, going by their age.
The only recent Game Center related code I have been able to find is "HelloGameKit" for WatchOS.
But I can't use this, because its in Swift, not Objective-C.
I know ObjectiveC is being phased out, but it would still be beneficial to have sample objC code for Game Center that is not using deprecated API calls.