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

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

as proved by Gowers.[8]

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

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

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) Value Reference

w(2; 3,3)

9

Chvátal [2]

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

Ramsey number
Graph coloring

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