Sure, I can help you write a test case generator for the problem "Escape of Lappin the Phantom Thief".

Here is a Python function to generate test cases for this problem:

```python
import random

def generate_test_case():
    # Define constraints
    n = random.randint(1, 5 * 10**4)
    m = random.randint(1, 5 * 10**4)
    k = random.randint(1, min(10**5, n * m))

    # Generate k unique robot coordinates
    robot_positions = set()
    while len(robot_positions) < k:
        x = random.randint(0, n - 1)
        y = random.randint(0, m - 1)
        robot_positions.add((x, y))

    # Convert set to list
    robot_positions = list(robot_positions)

    # Format the test case
    test_case = f"{n} {m} {k}\n"
    for x, y in robot_positions:
        test_case += f"{x} {y}\n"

    return test_case.strip()

# Example usage
print(generate_test_case())
```

### Explanation:

1. **Constraints**:
   - `n` and `m` are generated randomly within the range [1, 5 * 10^4].
   - `k` is generated randomly within the range [1, min(10^5, n * m)].

2. **Generating Unique Robot Coordinates**:
   - A set is used to ensure the robot coordinates are unique.
   - Random coordinates are generated and added to the set until its size reaches `k`.

3. **Formatting the Output**:
   - The test case is formatted as a string, with the first line containing `n`, `m`, and `k`.
   - Each subsequent line contains the coordinates `xi` and `yi` of the robots.

This generator creates random test cases that adhere to the given constraints, which can be used to test the algorithm for the problem "Escape of Lappin the Phantom Thief".