In the plane, $2018$ points are given such that all distances between them are different. For each point, mark the closest one of the remaining points. What is the minimal number of marked points?
1. **Understanding the problem**: We are given 2018 points in the plane such that all distances between them are different. For each point, we mark the closest one of the remaining points. We need to find the minimal number of marked points.

2. **Claim 1**: A point can be marked a maximum of 5 times.
   - **Reasoning**: Consider a point \( A \). If \( A \) is marked by another point \( B \), then \( A \) is the closest point to \( B \). Since all distances are different, the closest point to any point is unique. If \( A \) is marked by 6 different points, then \( A \) would be the closest point to each of these 6 points. However, in a plane, it is geometrically impossible for a single point to be the closest point to more than 5 other points due to the arrangement of distances and angles. This can be visualized by considering the Voronoi diagram, where each point has a region of influence, and no point can be the closest to more than 5 other points without violating the unique distance condition.

3. **Claim 2**: If point \( A \) has been marked 5 times and point \( A \) marks point \( B \) and point \( B \) marks \( A \), then \( B \) can be marked a maximum of 3 more times.
   - **Reasoning**: If \( A \) is marked by 5 points and marks \( B \), then \( B \) is the closest point to \( A \). If \( B \) also marks \( A \), then \( A \) is the closest point to \( B \). This mutual marking implies a strong geometric relationship between \( A \) and \( B \). Given that \( A \) is already marked by 5 points, the region around \( B \) is constrained. Therefore, \( B \) can be marked by at most 3 additional points without violating the unique distance condition.

4. **Claim 3**: If points \( A \) or \( B \) mark point \( C \), then point \( C \) can be marked a maximum of 2 more times.
   - **Reasoning**: If \( A \) or \( B \) marks \( C \), then \( C \) is the closest point to either \( A \) or \( B \). Given the constraints from the previous claims, the region around \( C \) is further constrained. Therefore, \( C \) can be marked by at most 2 additional points without violating the unique distance condition.

5. **Generalization**: For \( 9n + 2 \) points in the plane, the minimal number of marked points is \( 2m + 1 \).
   - **Reasoning**: This generalization is based on the pattern observed from the claims. For a given number of points, the minimal number of marked points follows a specific pattern. For \( 2018 \) points, which is of the form \( 9n + 2 \), the minimal number of marked points is \( 2m + 1 \).

The final answer is \( \boxed{ 2m + 1 } \).