Van der Waerden number

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).

Tables of Van der Waerden numbers
There are two cases in which the van der Waerden number W(r, k) is easy to compute: first, when the number of colors r is equal to 1, one has W(1, k) = k for any integer k, since one color produces only trivial colorings RRRRR...RRR (for the single color denoted R). Second, when the length k of the forced arithmetic progression is 2, one has W(r, 2) = r + 1, since one may construct a coloring that avoids arithmetic progressions of length 2 by using each color at most once, but using any color twice creates a length-2 arithmetic progression. (For example, for r = 3, the longest coloring that avoids an arithmetic progression of length 2 is RGB.) There are only seven other van der Waerden numbers that are known exactly. The table below gives exact values and bounds for values of W(r, k); values are taken from Rabung and Lotts except where otherwise noted.


 * {| class="wikitable"

! k\r ! 2 colors ! 3 colors ! 4 colors ! 5 colors ! 6 colors
 * 3
 * style="text-align:right;"| 9
 * style="text-align:right;"| 27
 * style="text-align:right;"| 76
 * style="text-align:right;"| >170
 * style="text-align:right;"| >225
 * 4
 * style="text-align:right;" | 35
 * style="text-align:right;"| 293
 * style="text-align:right;"| >1,048
 * style="text-align:right;"| >2,254
 * style="text-align:right;"| >9,778
 * 5
 * style="text-align:right;"| 178
 * style="text-align:right;"| >2,173
 * style="text-align:right;"| >17,705
 * style="text-align:right;"| >98,741
 * style="text-align:right;"| >98,748
 * 6
 * style="text-align:right;"| 1,132
 * style="text-align:right;"| >11,191
 * style="text-align:right;"| >157,209
 * style="text-align:right;"| >786,740
 * style="text-align:right;"| >1,555,549
 * 7
 * style="text-align:right;"| >3,703
 * style="text-align:right;"| >48,811
 * style="text-align:right;"| >2,284,751
 * style="text-align:right;"| >15,993,257
 * style="text-align:right;"| >111,952,799
 * 8
 * style="text-align:right;"| >11,495
 * style="text-align:right;"| >238,400
 * style="text-align:right;"| >12,288,155
 * style="text-align:right;"| >86,017,085
 * style="text-align:right;"| >602,119,595
 * 9
 * style="text-align:right;"| >41,265
 * style="text-align:right;"| >932,745
 * style="text-align:right;"| >139,847,085
 * style="text-align:right;"| >978,929,595
 * style="text-align:right;"| >6,852,507,165
 * 10
 * style="text-align:right;"| >103,474
 * style="text-align:right;"| >4,173,724
 * style="text-align:right;"| >1,189,640,578
 * style="text-align:right;"| >8,327,484,046
 * style="text-align:right;"| >58,292,388,322
 * 11
 * style="text-align:right;"| >193,941
 * style="text-align:right;"| >18,603,731
 * style="text-align:right;"| >3,464,368,083
 * style="text-align:right;"| >38,108,048,913
 * style="text-align:right;"| >419,188,538,043
 * }
 * style="text-align:right;"| >58,292,388,322
 * 11
 * style="text-align:right;"| >193,941
 * style="text-align:right;"| >18,603,731
 * style="text-align:right;"| >3,464,368,083
 * style="text-align:right;"| >38,108,048,913
 * style="text-align:right;"| >419,188,538,043
 * }
 * }

Some lower bound colorings computed using SAT approach by Marijn J.H. Heule can be found on github project page.

Van der Waerden numbers with r ≥ 2 are bounded above by
 * $$W(r,k)\le 2^{2^{r^{2^{2^{k+9}}}}}$$

as proved by Gowers.

For a prime number p, the 2-color van der Waerden number is bounded below by
 * $$p\cdot2^p\le W(2,p+1),$$

as proved by Berlekamp.

One sometimes also writes w(r; k1, k2, ..., kr) to mean the smallest number w such that any coloring of the integers {1, 2, ..., w} with r colors contains a progression of length ki of color i, for some i. Such numbers are called off-diagonal van der Waerden numbers. Thus W(r, k) = w(r; k, k, ..., k). Following is a list of some known van der Waerden numbers:

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.