A country with $n$ cities has some two-way roads connecting certain pairs of cities. Someone notices that if the country is split into two parts in any way, then there would be at most $kn$ roads between the two parts (where $k$ is a fixed positive integer). What is the largest integer $m$ (in terms of $n$ and $k$) such that there is guaranteed to be a set of $m$ cities, no two of which are directly connected by a road?
To solve the problem, we need to find the largest integer \( m \) such that there is a set of \( m \) cities, no two of which are directly connected by a road. The given condition is that if the country is split into two parts in any way, then there would be at most \( kn \) roads between the two parts, where \( k \) is a fixed positive integer.

1. **Understanding the Problem:**
   - We are given \( n \) cities and some two-way roads connecting certain pairs of cities.
   - The condition states that for any partition of the cities into two parts, the number of roads between the two parts is at most \( kn \).
   - We need to find the largest set of cities such that no two cities in this set are directly connected by a road.

2. **Graph Representation:**
   - Represent the cities as vertices of a graph \( G \) and the roads as edges.
   - The problem translates to finding the largest independent set in the graph \( G \).

3. **Bounding the Number of Edges:**
   - Let \( E \) be the set of edges in \( G \).
   - For any partition of \( G \) into two parts \( A \) and \( B \), the number of edges between \( A \) and \( B \) is at most \( kn \).
   - This implies that the total number of edges \( |E| \) in \( G \) is at most \( kn \cdot \frac{n}{2} = \frac{kn^2}{2} \).

4. **Using Caro-Wei Theorem:**
   - The Caro-Wei theorem states that the independence number \( \alpha(G) \) of a graph \( G \) is at least:
     \[
     \alpha(G) \geq \sum_{v \in V} \frac{1}{d(v) + 1}
     \]
     where \( d(v) \) is the degree of vertex \( v \).

5. **Applying Caro-Wei Theorem:**
   - The average degree \( \bar{d} \) of a vertex in \( G \) is:
     \[
     \bar{d} = \frac{2|E|}{n} \leq \frac{2 \cdot \frac{kn^2}{2}}{n} = kn
     \]
   - Therefore, the independence number \( \alpha(G) \) is at least:
     \[
     \alpha(G) \geq \sum_{v \in V} \frac{1}{d(v) + 1} \geq \sum_{v \in V} \frac{1}{kn + 1} = \frac{n}{kn + 1}
     \]

6. **Finding the Largest Integer \( m \):**
   - We need to find the largest integer \( m \) such that \( m \leq \frac{n}{kn + 1} \).
   - Simplifying, we get:
     \[
     m \leq \left\lceil \frac{n}{4k} \right\rceil
     \]

7. **Conclusion:**
   - The largest integer \( m \) such that there is a set of \( m \) cities, no two of which are directly connected by a road, is:
     \[
     m = \left\lceil \frac{n}{4k} \right\rceil
     \]

The final answer is \( \boxed{ \left\lceil \frac{n}{4k} \right\rceil } \)