Squeeze 3 tablespoons juice from limes into a large bowl. Add vegetable oil and mix.
Rinse cabbage, radishes, and cilantro. Finely shred enough cabbage to make 3 quarts. Thinly slice radishes to make about 1 cup. Chop enough cilantro to make about 1/2 cup. Add cabbage, radishes, and cilantro to dressing. Mix gently. Add salt and pepper to taste.