Preheat the oven to 350 degrees. Brush the cake pans with butter. Line each with parchment paper; butter the parchment, and dust with cocoa powder, tapping out any excess; set aside. Whisk together the flour, salt, and cocoa powder in a medium bowl; set aside.
Beat the sugar and oil on medium speed in a mixer bowl until combined. Add the eggs one at a time, mixing well after each addition. Mix in the food coloring and vanilla. Add the flour mixture in three batches, alternating with the buttermilk and beginning and ending with flour, mixing well after each addition. Scrape down the sides of the bowl as needed.
Stir together the baking soda and vinegar in a small bowl. Add the baking-soda mixture to the batter, and mix on medium speed for 10 seconds. Pour the batter into the prepared pans. Bake until a cake tester inserted into the centers comes out clean. Let cool completely in the pans on wire racks.
Approximate batter amounts and baking times for 3-inch-deep round pans: 4-inch: 1 1⁄2 cups, 25 minutes; 6-inch: 3 1⁄4 cups, 45 minutes; 8-inch: 5 1⁄2 cups, 1 hour 5 minutes; 10-inch: 9 cups, 1 hour 20 minutes; 12-inch: 12 1⁄4 cups, 1 hour 25 minutes.
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