Just replace LIKE with CONTAINS in your format string. For example, given this array:
NSArray *dogs = @[@{@"name" : @"Fido",
@"toys" : @[@"Ball", @"Kong"]},
@{@"name" : @"Rover",
@"toys" : @[@"Ball", @"Rope"]},
@{@"name" : @"Spot",
@"toys" : @[@"Rope", @"Kong"]}];
...the following predicate can be used obtain a filtered array containing only the dictionaries where the value for the key toy is an array that contains the string Kong.
NSPredicate *predicate = [NSPredicate predicateWithFormat:@"%K CONTAINS[cd] %@", @"toys", @"Kong"];