Van der Waerden's Theorem
Van der Waerden's theorem states that for positive integers and there exists an integer such that any -coloring of the integers contains a monochromatic arithmetic progression (MAP) of length . The current best-known bounds on are large power towers; very few specific values are known. After seeding a random coloring, you can click a number to cycle through other colors. See if you can find a 3-coloring of 1 through 26 with no MAP of length 3, thus showing . What about extremal colorings for , , and ?