 For a prime number "p", the 2-color van der Waerden number is bounded below by
as proved by Berlekamp.One sometimes also writes "w"(r; "k"1, "k"2, ..., "k""r") to mean the smallest number "w" such that any coloring of the integers {1, 2, ..., "w"} with "r" colors contains a progression of length "k""i" of color "i", for some "i".Such numbers are called "off-diagonal van der Waerden numbers".