Put the potatoes in a large saucepan or Dutch oven. Pour in enough cold water to cover and add a small handful of salt. Bring to a boil over high heat. Adjust the heat so the water is simmering and cook until the potatoes can be easily pierced with a paring knife, about 10 minutes.
While the potatoes are cooking, start making the eggs: Put the eggs in a separate saucepan and pour in enough cold water to cover. Bring to a boil over high heat, then adjust the heat to medium-low. Cook the eggs for exactly 10 minutes. Drain the hot water and cool the eggs in the pan under cold running water for several minutes. Peel, and set aside.
When the potatoes are tender, drain them and put them in a serving bowl. Immediately sprinkle the vinegar over them and set aside to cool. Don't stir the potatoes too much while still hot, as they tend to fall apart and look unattractive.
Once the potatoes are cool, add the olive oil and chives and stir gently to coat the potatoes with oil. Season the salad, to taste, with salt and pepper and toss again. Cut the eggs into quarters and decorate the top of the salad. Sprinkle with diced red pepper, if using. Serve at room temperature.