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:
| 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.
See also
Ramsey number
Graph coloring
References
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Kouril, Michal; Paul, Jerome L. (2008). "The Van der Waerden Number W(2,6) is 1132". Experimental Mathematics. 17 (1): 53–61. doi:10.1080/10586458.2008.10129025. MR 2410115.
"Daniel Monroe, Van Der Waerden Numbers". vdwnumbers.org. Retrieved 2015-09-19.
Gowers, Timothy (2001). "A new proof of Szemerédi's theorem". Geom. Funct. Anal. 11 (3): 465–588. doi:10.1007/s00039-001-0332-9. MR 1844079.
Berlekamp, E. (1968). "A construction for partitions which avoid long arithmetic progressions". Canadian Mathematical Bulletin. 11 (3): 409–414. doi:10.4153/CMB-1968-047-7. MR 0232743.
Landman, Bruce; Robertson, Aaron; Culver, Clay (2005). "Some New Exact van der Waerden Numbers" (PDF). Integers. 5 (2): A10. MR 2192088.
Kouril, Michal (2006). A Backtracking Framework for Beowulf Clusters with an Extension to Multi-Cluster Computation and Sat Benchmark Problem Implementation (Ph.D. thesis). University of Cincinnati.
Ahmed, Tanbir (2010). "Two new van der Waerden numbers w(2;3,17) and w(2;3,18)". Integers. 10 (4): 369–377. doi:10.1515/integ.2010.032. MR 2684128.
Ahmed, Tanbir; Kullmann, Oliver; Snevily, Hunter (2014). "On the van der Waerden numbers w(2;3,t)". Discrete Appl. Math. 174 (2014): 27–51. arXiv:1102.5433. doi:10.1016/j.dam.2014.05.007. MR 3215454.
Kouril, Michal (2015). "Leveraging FPGA clusters for SAT computations". Parallel Computing: On the Road to Exascale: 525–532.
Beeler, Michael D. (1983). "A new van der Waerden number". Discrete Applied Mathematics. 6 (2): 207. doi:10.1016/0166-218x(83)90073-2. MR 0707027.
Ahmed, Tanbir (2012). "On computation of exact van der Waerden numbers". Integers. 12 (3): 417–425. doi:10.1515/integ.2011.112. MR 2955523.
Ahmed, Tanbir (2013). "Some More Van der Waerden numbers". Journal of Integer Sequences. 16 (4): 13.4.4. MR 3056628.
Brown, T. C. (1974). "Some new van der Waerden numbers (preliminary report)". Notices of the American Mathematical Society. 21: A-432.
Ahmed, Tanbir (2009). "Some new van der Waerden numbers and some van der Waerden-type numbers". Integers. 9: A6. doi:10.1515/integ.2009.007. MR 2506138.
Schweitzer, Pascal (2009). Problems of Unknown Complexity, Graph isomorphism and Ramsey theoretic numbers (Ph.D. thesis). U. des Saarlandes.
Shelah, Saharon (1988). "Primitive recursive bounds for van der Waerden numbers". Journal of the American Mathematical Society 1 (3): 683–697. doi:10.2307/1990952. JSTOR 1990952. MR 0929498.
Further reading
Landman, Bruce M.; Robertson, Aaron (2014). Ramsey Theory on the Integers. Student Mathematical Library. 73 (Second ed.). Providence, RI: American Mathematical Society. doi:10.1090/stml/073. ISBN 978-0-8218-9867-3. MR 3243507.
Herwig, P. R.; Heule, M. J. H.; van Lambalgen, P. M.; van Maaren, H. (2007). "A New Method to Construct Lower Bounds for Van der Waerden Numbers". The Electronic Journal of Combinatorics. 14 (1). MR 2285810.
External links
O'Bryant, Kevin and Weisstein, Eric W. "Van der Waerden Number". MathWorld.
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