I want to remove Facebook Birthdays.

You have Facebook account on the iPhone Settings app. Then iOS fetches all birthday of friends to your calendar. Sometimes, it messes up your calendar. 

To turn off the Facebook Events and Birthdays, you need to turn off both Calendar and Contacts in the Facebook settings.

The iPhone calendar retrieves Facebook events only and the Contacts retrieves Facebook friends contact list only and if the contacts has birthday data(unfortunately, almost contacts on Facebook have birthdays), it will show the birthday on the Calendar

So you can selectively turn off one of them if you want.

If you turn off Calendar, Facebook Events will be removed

If you turn off Contacts, Facebook birthdays will be removed. 

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