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Peel potatoes and cut into 1/2-inch cubes. Place in a large pot and add water to cover; season with 1 teaspoon salt. Bring to a boil over high heat. Reduce the heat and simmer until very tender, 15 to 20 minutes. Drain well. While still warm, transfer the potatoes to a baking sheet and drizzle with vinegar. Set aside to cool to room temperature.
Meanwhile, place eggs in a medium saucepan and add water to cover by 1 inch. Bring to a simmer over medium-high heat. Reduce heat to low and cook at the barest simmer for 10 minutes. Remove from heat, pour out the hot water and cover the eggs with ice-cold water. Let stand until cool enough to handle before peeling.
Once the eggs have cooled, remove the shells by tapping each egg gently on the counter or sink all over to crackle it. Roll an egg between your hands to loosen the shell. Peel, starting at the large end, while holding the egg under cold running water; this facilitates peeling. Grate the eggs through the large holes on a box grater or finely chop them.
Combine mayonnaise, celery, onion, pickle relish, the remaining 1/2 teaspoon salt and pepper in a large bowl. Add the cooled potatoes and grated eggs; stir to combine. Serve at room temperature or chilled.