Remove the shells from the hard boiled eggs and slice each egg in half. Remove the yolks from the egg halves and place in a bowl. Set the whites aside.
Get enough glasses or mugs for the number of colors you want to use. Fill the mugs or glasses 2/3 of the way full with cold water. Add 3 drops of desired food coloring along with 1 tsp vinegar to each mug or glass.
Place egg whites in the mugs and allow to sit in the food coloring until desired color is reached. You can add more then one egg white in the mug at a time. Remove whites from the dye and drain on a plate lined with paper towels.
Take the reserved yolks and mash them with a fork. Then mix in the 1/2 cup of mayonnaise with the mashed yolk. Pipe or spoon the yolk mix into the colored egg whites.