There is one webpage with instructions on how to install Ubuntu on a Macbook Air 6,1 model:

http://uli-heller.github.io/blog/2013/06/24/macbook-air/ (in German - I'll provide an English translation on demand, use "bcmwl" from 13.10 to get WLAN up and running)

http://www.miek.nl/blog/archives/2013/08/31/macbook_air_61_2013_model_with_ubuntu/index.html

http://mattoncloud.org/2013/07/18/fedora-19-on-a-macbook-air-2013-model/ (Fedora install, but still has useful information)