Place potato cubes into a small pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, 5 to 10 minutes. Drain and let cool, about 10 minutes.
Combine cooled potato, cabbage, pomegranate seeds, and carrot in a large bowl.
Whisk olive oil, lemon juice, garlic, salt, brown sugar, and red pepper flakes in a small bowl until well-mixed. Poured over potato mixture and toss to combine.