The airline is incorrect. Any part of a calendar day counts as one day. If you enter on the 11th, you must leave on or before the 17th.
This is unfortunately not explicit in the Schengen Visa Code, but the Schengen Borders Code describes how days of presence are counted for the 90/180 rule, and it is safe to assume that the same principle is applied to counting days for the "duration of stay" limitation of a Schengen visa:
the date of entry shall be considered as the first day of stay on the territory of the Member States and the date of exit shall be considered as the last day of stay on the territory of the Member States.