In a country there are $n\geq 2$ cities. Any two cities has exactly one two-way airway. The government wants to license several airlines to take charge of these airways with such following conditions:

i) Every airway can be licensed to exactly one airline.
ii) By choosing one arbitrary airline, we can move from a city to any other cities, using only flights from this airline.

What is the maximum number of airlines that the government can license to satisfy all of these conditions?
1. **Restate the problem in graph-theoretic terms:**
   We need to color each edge of the complete graph on \( n \) vertices (i.e., \( K_n \)) with colors \( C_1, C_2, \dots, C_k \) such that:
   - Each edge is colored by only one color.
   - For each color \( C_i \), there exists a monochromatic path of color \( C_i \) passing through all vertices of \( K_n \).

2. **Determine the maximum possible value of \( k \) in terms of \( n \):**
   - For the second condition to be true, there must exist at least \( n-1 \) edges of color \( C_i \) for all \( i \in [k] \).
   - Therefore, we have:
     \[
     k(n-1) \leq \binom{n}{2}
     \]
   - Simplifying the binomial coefficient:
     \[
     \binom{n}{2} = \frac{n(n-1)}{2}
     \]
   - Thus:
     \[
     k(n-1) \leq \frac{n(n-1)}{2} \Rightarrow k \leq \frac{n}{2}
     \]

3. **Show that the claimed value \( k = \left\lfloor \frac{n}{2} \right\rfloor \) is achievable:**
   - **Case 1: \( n \) is even.**
     - Let \( n = 2m \). We prove the result by induction on \( m \).
     - **Base case:** For \( m = 1 \), \( n = 2 \), and \( k = 1 \). The complete graph \( K_2 \) has one edge, which can be colored with one color.
     - **Inductive step:** Assume the result is true for \( m-1 \). Consider the graph \( K_{2m} \). Label the vertices as \( v_1, v_2, \dots, v_{2m} \).
       - By the induction hypothesis, we can color the complete graph on vertices \( v_1, v_2, \dots, v_{2m-2} \) with colors \( C_1, C_2, \dots, C_{m-1} \) such that the given conditions are satisfied.
       - Now consider the vertices \( v_{2m-1} \) and \( v_{2m} \). Color the edges \( v_{2m-1}v_{2j-1} \) and \( v_{2m}v_{2j} \) with color \( C_j \) for all \( j \in [m-1] \).
       - Also, color the edges \( v_{2m-1}v_{2j} \) and \( v_{2m}v_{2j-1} \) with a new color \( C_m \) for all \( j \in [m] \).
       - This coloring satisfies the given conditions, completing the induction.

   - **Case 2: \( n \) is odd.**
     - Remove one vertex from the graph, making it even. Color the remaining complete graph with an even number of vertices as shown above.
     - Add the removed vertex back in and color the edges coming out of that vertex with all the colors (in some order) present in our coloring of the even complete graph.
     - This ensures that the conditions are satisfied for odd \( n \) as well.

Thus, the maximum number of airlines that the government can license is \( \left\lfloor \frac{n}{2} \right\rfloor \).

The final answer is \( \boxed{ \left\lfloor \frac{n}{2} \right\rfloor } \)