Name conflicts are almost never an issue, unless you run into an extremely uptight agent. Unfortunately, in that instance, they are legally in the right and can refuse to honor the ticket. That said, my name is of the form:

Robert Jonathan Michael Smith

and I've flown many millions of international miles with tickets that read Jon Smith. Checking in this morning, in fact, I handed the UA agent my passport and she asked "You go by Jon?" I nodded, and she checked me in.

While I would never recommend booking international tickets in anything other than your full legal name (do as I say, not as I do I wouldn't loose a moment's sleep over the OP's issue, let alone pay UA money to fix their mistake!

(To the previous poster, the TSA is not the problem here. The legal issue is with immigration, so you'll certainly need to make sure visas (if any) do fully match the passport. The airline only cares in as much as they want to prevent ticket fraud and make sure that you will pass immigration on the other end.)