Sure, I can help you build a test case generator for "Problem K: Escape of Lappin the Phantom Thief." Here is a Python script to generate random test cases based on the problem constraints:

```python
import random

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

# Example usage
n, m, k, coordinates = tcgen()
print(n, m, k)
for (x, y) in coordinates:
    print(x, y)
```

### Explanation

1. **Constraints**:
   - \( n \) and \( m \) are generated randomly within the range [1, 50,000].
   - \( k \) is generated randomly within the range [1, min(100,000, \( n \times m \))].

2. **Generating Robot Coordinates**:
   - A set is used to ensure all coordinates are unique.
   - Random coordinates (xi, yi) are generated until there are \( k \) unique coordinates.

3. **Output**:
   - The first line contains \( n \), \( m \), and \( k \).
   - The following \( k \) lines contain the coordinates of the robots.

### Sample Output

Here’s an example of how the output might look when you run the test case generator:

```
20438 38570 5
3145 28273
18230 10446
14786 16991
12120 327
783 3778
```

You can use the `tcgen` function to generate multiple test cases for thorough testing of your solution.