If your MacBook Air came with Mavericks installed, you can reinstall by booting to the Apple servers via the internet by restarting and hold the Command+Option+R keys. The boot will take longer but you can then install Mac OS X.
A basic question is, why did you format the internal storage? You should never do that without making a backup on another hard drive so you have a path to restore the system.
If you are not able to get an internet connection, try taking the MacBook Air to an Apple store genius bar and see if they can install Mac OS X from their system.