Question 27
Between any two cities of country $A$ consisting of $2011$ cities and country $B$ consisting of $2011$ cities there is a unique direct two way flight organized by some airway company. For each given city there are at most $19$ different airway companies operating flights related to this city. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged there are $k$ cities connected (not necessarily directly) only by the flights of some fixed airway company.