Van Der Waerden Number

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Van der Waerden's theorem states that for any positive integers r and k there exists a positive integer N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression all of the same color. The smallest such N is the van der Waerden number W(r,k). Van der Waerden numbers are primitive recursive, as proved by Shelah; in fact he proved that they are (at most) on the fifth level mathcal{E}^5 of the Grzegorczyk hierarchy. W(1,k)=k for any integer k, since one color produces only trivial colorings RRRRR...RRR (for the single color denoted R). W(r,2)=r+1, since we may construct a coloring that avoids arithmetic progressions of length 2 by using each color at most once, but once we use a color twice, a length 2 arithmetic progression is formed (e.g., for r=3, the longest coloring we can get that avoids an arithmetic progression of length 2 is RGB).