4.84 A certain country has 21 cities, and several airlines are responsible for air transport between them. Each airline operates direct flights (non-stop and direct) between 5 cities (there can be multiple airlines operating flights between two cities), and there is at least one direct flight between any two cities. How many airlines are needed at a minimum?
[Solution 1] To ensure that this country has a satisfactory aviation network, with at least one flight between every two cities, there should be at least $C_{21}^{2}=210$ flights. Each airline operates 10 flights, so there should be at least 21 airlines.

In the graph, we take the 21 cities as the vertices of a regular 21-sided polygon. The graph drawn represents the flight routes of the first airline, which connects 5 cities with the numbers $1,3,8,9,12$. The advantage of this selection is that the 10 lines connecting the 5 points (measured by the length of the shorter arc) have lengths of $1,2,3,4,5,6,7,8,9,10$, all of which are different. Therefore, when the numbers are rotated to derive the city number groups served by the other 20 airlines, the resulting 210 lines are all unique, ensuring that there is exactly one direct flight between every two cities. Thus, the minimum number of airlines required is 21.

[Solution 2] As in Solution 1, it can be proven that at least 21 airlines are needed.
Below, we use the "dictionary order method" to list the city numbers operated by each of the 21 airlines, ensuring that any two groups of numbers share exactly one repetition. This ensures that the flight routes operated by them are all unique. This guarantees that the 210 routes are exactly one between each pair of cities. The 21 groups of numbers are specifically arranged as follows:
$$
\begin{array}{l}
(1,2,3,4,5),(1,6,7,8,9), \\
(1,10,11,12,13),(1,14,15,16,17), \\
(1,18,19,20,21),(2,6,10,14,18), \\
(2,7,11,15,19),(2,8,12,16,20), \\
(2,9,13,17,21),(3,6,11,16,21), \\
(3,7,10,17,20),(3,8,13,14,19), \\
(3,9,12,15,18),(4,6,12,17,19), \\
(4,7,13,16,18),(4,8,10,15,21), \\
(4,9,11,14,20),(5,6,13,15,20), \\
(5,7,12,14,21),(5,8,11,17,18), \\
(5,9,10,16,19).
\end{array}
$$

It is evident that to meet the requirements of the problem, at least 21 airlines are needed.