# 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.[1]

k\r 2 colors 3 colors 4 colors 5 colors 6 colors
3 9[2] 27[2]   76[3]   >170   >223
4 35[2] 293[4]   >1,048   >2,254   >9,778
5 178[5] >2,173   >17,705   >98,740   >98,748
6 1,132[6] >11,191   >91,331   >540,025   >816,981
7 >3,703   >48,811   >420,217   >1,381,687   >7,465,909
8 >11,495   >238,400   >2,388,317   >10,743,258   >57,445,718
9 >41,265   >932,745   >10,898,729   >79,706,009   >458,062,329[7]
10 >103,474   >4,173,724   >76,049,218   >542,694,970[7] >2,615,305,384[7]
11 >193,941   >18,603,731   >305,513,57[7] >2,967,283,511[7] >3,004,668,671[7]

Van der Waerden numbers with r ≥ 2 are bounded above by

${\displaystyle W(r,k)\leq 2^{2^{r^{2^{2^{k+9}}}}}}$

as proved by Gowers.[8]

For a prime number p, the 2-color van der Waerden number is bounded below by

${\displaystyle p\cdot 2^{p}\leq W(2,p+1),}$

as proved by Berlekamp.[9]

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:

Known van der Waerden numbers
w(r;k1, k2, …, kr)ValueReference

w(2; 3,3)

9

Chvátal [2]

w(2; 3,4)18Chvátal [2]
w(2; 3,5)22Chvátal [2]
w(2; 3,6)32Chvátal [2]
w(2; 3,7)46Chvátal [2]
w(2; 3,8)58Beeler and O'Neil [3]
w(2; 3,9)77Beeler and O'Neil [3]
w(2; 3,10)97Beeler and O'Neil [3]
w(2; 3,11)114Landman, Robertson, and Culver [10]
w(2; 3,12)135Landman, Robertson, and Culver [10]
w(2; 3,13)160Landman, Robertson, and Culver [10]
w(2; 3,14)186Kouril [11]
w(2; 3,15)218Kouril [11]
w(2; 3,16)238Kouril [11]
w(2; 3,17)279Ahmed [12]
w(2; 3,18)312Ahmed [12]
w(2; 3,19)349Ahmed, Kullmann, and Snevily [13]
w(2; 3,20)389Ahmed, Kullmann, and Snevily [13] (conjectured); Kouril [14] (verified)
w(2; 4,4)35Chvátal [2]
w(2; 4,5)55Chvátal [2]
w(2; 4,6)73Beeler and O'Neil [3]
w(2; 4,7)109Beeler [15]
w(2; 4,8)146Kouril [11]
w(2; 4,9)309Ahmed [16]
w(2; 5,5)178Stevens and Shantaram [5]
w(2; 5,6)206Kouril [11]
w(2; 5,7)260Ahmed [17]
w(2; 6,6)1132Kouril and Paul [6]
w(3; 2, 3, 3)14Brown [18]
w(3; 2, 3, 4)21Brown [18]
w(3; 2, 3, 5)32Brown [18]
w(3; 2, 3, 6)40Brown [18]
w(3; 2, 3, 7)55Landman, Robertson, and Culver [10]
w(3; 2, 3, 8)72Kouril [11]
w(3; 2, 3, 9)90Ahmed [19]
w(3; 2, 3, 10)108Ahmed [19]
w(3; 2, 3, 11)129Ahmed [19]
w(3; 2, 3, 12)150Ahmed [19]
w(3; 2, 3, 13)171Ahmed [19]
w(3; 2, 3, 14)202Kouril [4]
w(3; 2, 4, 4)40Brown [18]
w(3; 2, 4, 5)71Brown [18]
w(3; 2, 4, 6)83Landman, Robertson, and Culver [10]
w(3; 2, 4, 7)119Kouril [11]
w(3; 2, 4, 8)157Kouril [4]
w(3; 2, 5, 5)180Ahmed [19]
w(3; 2, 5, 6)246Kouril [4]
w(3; 3, 3, 3)27Chvátal [2]
w(3; 3, 3, 4)51Beeler and O'Neil [3]
w(3; 3, 3, 5)80Landman, Robertson, and Culver [10]
w(3; 3, 3, 6)107Ahmed [16]
w(3; 3, 4, 4)89Landman, Robertson, and Culver [10]
w(3; 4, 4, 4)293Kouril [4]
w(4; 2, 2, 3, 3)17Brown [18]
w(4; 2, 2, 3, 4)25Brown [18]
w(4; 2, 2, 3, 5)43Brown [18]
w(4; 2, 2, 3, 6)48Landman, Robertson, and Culver [10]
w(4; 2, 2, 3, 7)65Landman, Robertson, and Culver [10]
w(4; 2, 2, 3, 8)83Ahmed [19]
w(4; 2, 2, 3, 9)99Ahmed [19]
w(4; 2, 2, 3, 10)119Ahmed [19]
w(4; 2, 2, 3, 11)141Schweitzer [20]
w(4; 2, 2, 3, 12)163Kouril [14]
w(4; 2, 2, 4, 4)53Brown [18]
w(4; 2, 2, 4, 5)75Ahmed [19]
w(4; 2, 2, 4, 6)93Ahmed [19]
w(4; 2, 2, 4, 7)143Kouril [4]
w(4; 2, 3, 3, 3)40Brown [18]
w(4; 2, 3, 3, 4)60Landman, Robertson, and Culver [10]
w(4; 2, 3, 3, 5)86Ahmed [19]
w(4; 2, 3, 3, 6)115Kouril [14]
w(4; 3, 3, 3, 3)76Beeler and O'Neil [3]
w(5; 2, 2, 2, 3, 3)20Landman, Robertson, and Culver [10]
w(5; 2, 2, 2, 3, 4)29Ahmed [19]
w(5; 2, 2, 2, 3, 5)44Ahmed [19]
w(5; 2, 2, 2, 3, 6)56Ahmed [19]
w(5; 2, 2, 2, 3, 7)72Ahmed [19]
w(5; 2, 2, 2, 3, 8)88Ahmed [19]
w(5; 2, 2, 2, 3, 9)107Kouril [4]
w(5; 2, 2, 2, 4, 4)54Ahmed [19]
w(5; 2, 2, 2, 4, 5)79Ahmed [19]
w(5; 2, 2, 2, 4, 6)101Kouril [4]
w(5; 2, 2, 3, 3, 3)41Landman, Robertson, and Culver [10]
w(5; 2, 2, 3, 3, 4)63Ahmed [19]
w(5; 2, 2, 3, 3, 5)95Kouril [14]
w(6; 2, 2, 2, 2, 3, 3)21Ahmed [19]
w(6; 2, 2, 2, 2, 3, 4)33Ahmed [19]
w(6; 2, 2, 2, 2, 3, 5)50Ahmed [19]
w(6; 2, 2, 2, 2, 3, 6)60Ahmed [19]
w(6; 2, 2, 2, 2, 4, 4)56Ahmed [19]
w(6; 2, 2, 2, 3, 3, 3)42Ahmed [19]
w(7; 2, 2, 2, 2, 2, 3, 3)24Ahmed [19]
w(7; 2, 2, 2, 2, 2, 3, 4)36Ahmed [19]
w(7; 2, 2, 2, 2, 2, 3, 5)55Ahmed [16]
w(7; 2, 2, 2, 2, 2, 3, 6)65Ahmed [17]
w(7; 2, 2, 2, 2, 2, 4, 4)66Ahmed [17]
w(7; 2, 2, 2, 2, 3, 3, 3)45Ahmed [17]
w(8; 2, 2, 2, 2, 2, 2, 3, 3)25Ahmed [19]
w(8; 2, 2, 2, 2, 2, 2, 3, 4)40Ahmed [16]
w(8; 2, 2, 2, 2, 2, 2, 3, 5)61Ahmed [17]
w(8; 2, 2, 2, 2, 2, 2, 3, 6)71Ahmed [17]
w(8; 2, 2, 2, 2, 2, 2, 4, 4)67Ahmed [17]
w(8; 2, 2, 2, 2, 2, 3, 3, 3)49Ahmed [17]
w(9; 2, 2, 2, 2, 2, 2, 2, 3, 3)28Ahmed [19]
w(9; 2, 2, 2, 2, 2, 2, 2, 3, 4)42Ahmed [17]
w(9; 2, 2, 2, 2, 2, 2, 2, 3, 5)65Ahmed [17]
w(9; 2, 2, 2, 2, 2, 2, 3, 3, 3)52Ahmed [17]
w(10; 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)31Ahmed [17]
w(10; 2, 2, 2, 2, 2, 2, 2, 2, 3, 4)45Ahmed [17]
w(10; 2, 2, 2, 2, 2, 2, 2, 2, 3, 5)70Ahmed [17]
w(11; 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)33Ahmed [17]
w(11; 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4)48Ahmed [17]
w(12; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)35Ahmed [17]
w(12; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4)52Ahmed [17]
w(13; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)37Ahmed [17]
w(13; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4)55Ahmed [17]
w(14; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)39Ahmed [17]
w(15; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)42Ahmed [17]
w(16; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)44Ahmed [17]
w(17; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)46Ahmed [17]
w(18; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)48Ahmed [17]
w(19; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)50Ahmed [17]
w(20; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3)51Ahmed [17]

Van der Waerden numbers are primitive recursive, as proved by Shelah;[21] in fact he proved that they are (at most) on the fifth level ${\displaystyle {\mathcal {E}}^{5}}$ of the Grzegorczyk hierarchy.

## References

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