RIOPLATENSE 2011 Problem 3

Let $M$ be a map made of several cities linked to each other by flights. We say that there is a route between two cities if there is a nonstop flight linking these two cities. For each city to the $M$ denote by $M_a$ the map formed by the cities that have a route to and routes linking these cities together ( to not part of $M_a$ ). The cities of $M_a$ are divided into two sets so that the number of routes linking cities of different sets is maximum; we call this number the cut of $M_a$ . Suppose that for every cut of $M_a$ , it is strictly less than two thirds of the number of routes $M_a$ . Show that for any coloring of the $M$ routes with two colors there are three cities of $M$ joined by three routes of the same color.