25.6. (USSR, 81; USA, 81). In a certain country, any two cities are directly connected by one of the following means of transportation: bus, train, or airplane. It is known that there is no city provided with all three types of transportation, and at the same time, there do not exist three cities such that any two of them are connected by the same means of transportation. Find the largest possible number of cities in this country.
25.6. Suppose there are five cities connected in the manner indicated in the problem. First, let's prove that no city has three lines of the same type of transport leading out of it. Let city \( A \) be connected to cities \( B, C \), and \( D \), for example, by airplane. Then, according to the condition, no pair of cities \( B, C \), and \( D \) can be connected by airplane. Suppose \( B \) and \( C \) are connected, for example, by train. Cities \( C \) and \( D \) cannot be connected by bus, as otherwise city \( C \) would have all three types of transport. Therefore, \( C \) and \( D \) are connected by train. For the same reasons, cities \( B \) and \( D \) are also connected by train. We have obtained that \( B, C \), and \( D \) are pairwise connected by train. Contradiction. Thus, from each city, two transport lines of one type and two transport lines of another type lead out. Then each city is served by exactly two types of transport. Therefore, at least one type of transport serves no more than three cities (otherwise, there would be at least \( 4 \cdot 3 / 2 = 6 \) cities in total). If it serves exactly two cities, then from each of these cities, only one transport line of this type leads out, which is impossible. If it serves exactly three cities, then they must be pairwise connected by this type of transport, which is also impossible. Thus, we have proved that the condition of the problem is unfulfillable for five cities. It is even more unfulfillable for a larger number of cities. If, however, we consider four cities \( A, B \), \( C \), and \( D \), which are connected as follows: \( A \) with \( B \) by train, \( C \) with \( D \) by bus, and all other pairs by airplane, then all conditions of the problem will be met. Therefore, the maximum number of cities is 4.