1. Whisk together vinegar, oil, juice, pepper flakes, and sugar in a large bowl with salt and pepper to taste.
2. Peel and cut jicama into 1/8-inch-thick slices, then cut into matchsticks to measure 2 cups. Add to bowl with dressing. Cut 1/8-inch-thick slices away from sides of mango, then cut crosswise into matchsticks to measure about 2-1/2 cups and add to bowl. Halve papaya and scoop out seeds. Cut papaya lengthwise into 1/8-inch-thick slices, then cut into matchsticks to measure about 1 cup and add to bowl. Add remaining ingredients to bowl and gently toss to combine. Season with salt and pepper to taste.