Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.
1. **Understanding the Problem:**
   We are given a \( n \times n \) board filled with integers from \( 1 \) to \( n^2 \). In each row, the \( m \) largest numbers are colored red, and in each column, the \( m \) largest numbers are colored blue. We need to find the minimum number of cells that are colored both red and blue.

2. **Initial Observation:**
   Since \( m \) largest numbers in each row are colored red and \( m \) largest numbers in each column are colored blue, we need to find the overlap between these two sets of numbers.

3. **Strategy:**
   To minimize the overlap, we need to consider the worst-case scenario where the overlap is minimized. However, since we are dealing with the largest numbers in each row and column, there will be some inevitable overlap.

4. **Counting Overlaps:**
   Let's consider the \( m \) largest numbers in each row. There are \( n \) rows, so there are \( n \times m \) red cells in total. Similarly, there are \( n \times m \) blue cells in total.

5. **Pigeonhole Principle:**
   By the pigeonhole principle, since there are \( n \times m \) red cells and \( n \times m \) blue cells in a \( n \times n \) board, and each row and column can have at most \( n \) cells, there must be some overlap.

6. **Minimum Overlap Calculation:**
   To find the minimum number of cells that are both red and blue, consider the following:
   - Each row has \( m \) red cells.
   - Each column has \( m \) blue cells.
   - To minimize the overlap, we need to distribute the red and blue cells as evenly as possible.

7. **Conclusion:**
   The minimum number of cells that are both red and blue is \( m^2 \). This is because, in the worst-case scenario, each of the \( m \) largest numbers in \( m \) rows will overlap with the \( m \) largest numbers in \( m \) columns.

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