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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

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

Some lower bound colorings computed using SAT approach by Marijn J.H. Heule [6] can be found on github project page.

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

 

as proved by Gowers.[9]

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

 

as proved by Berlekamp.[10]

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

Van der Waerden numbers are primitive recursive, as proved by Shelah;[23] in fact he proved that they are (at most) on the fifth level   of the Grzegorczyk hierarchy.

See also

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References

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  1. ^ Rabung, John; Lotts, Mark (2012). "Improving the use of cyclic zippers in finding lower bounds for van der Waerden numbers". Electron. J. Combin. 19 (2). doi:10.37236/2363. MR 2928650.
  2. ^ a b c d e f g h i j k Chvátal, Vašek (1970). "Some unknown van der Waerden numbers". In Guy, Richard; Hanani, Haim; Sauer, Norbert; et al. (eds.). Combinatorial Structures and Their Applications. New York: Gordon and Breach. pp. 31–33. MR 0266891.
  3. ^ a b c d e f g Beeler, Michael D.; O'Neil, Patrick E. (1979). "Some new van der Waerden numbers". Discrete Mathematics. 28 (2): 135–146. doi:10.1016/0012-365x(79)90090-6. MR 0546646.
  4. ^ a b c d e f g h Kouril, Michal (2012). "Computing the van der Waerden number W(3,4)=293". Integers. 12: A46. MR 3083419.
  5. ^ a b Stevens, Richard S.; Shantaram, R. (1978). "Computer-generated van der Waerden partitions". Mathematics of Computation. 32 (142): 635–636. doi:10.1090/s0025-5718-1978-0491468-x. MR 0491468.
  6. ^ a b Heule, MarijnJ (2017). "Avoiding triples in arithmetic progression" (PDF). Journal of Combinatorics. 8: 391–422.
  7. ^ a b 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. S2CID 1696473.
  8. ^ a b c d e f g h i j k l m n o p q r Monroe, Daniel (2019). "New Lower Bounds for van der Waerden Numbers Using Distributed Computing". arXiv:1603.03301 [math.CO].
  9. ^ Gowers, Timothy (2001). "A new proof of Szemerédi's theorem" (PDF). Geom. Funct. Anal. 11 (3): 465–588. doi:10.1007/s00039-001-0332-9. MR 1844079. S2CID 124324198.
  10. ^ 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.
  11. ^ a b c d e f g h i j k l Landman, Bruce; Robertson, Aaron; Culver, Clay (2005). "Some New Exact van der Waerden Numbers" (PDF). Integers. 5 (2): A10. MR 2192088.
  12. ^ a b c d e f g 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.
  13. ^ a b 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. S2CID 124272560.
  14. ^ a b Ahmed, Tanbir; Kullmann, Oliver; Snevily, Hunter (2014). "On the van der Waerden numbers w(2;3,t)". Discrete Applied Mathematics. 174 (2014): 27–51. arXiv:1102.5433. doi:10.1016/j.dam.2014.05.007. MR 3215454.
  15. ^ a b c d Kouril, Michal (2015). "Leveraging FPGA clusters for SAT computations". Parallel Computing: On the Road to Exascale: 525–532.
  16. ^ 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.
  17. ^ a b c d Ahmed, Tanbir (2012). "On computation of exact van der Waerden numbers". Integers. 12 (3): 417–425. doi:10.1515/integ.2011.112. MR 2955523. S2CID 11811448.
  18. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa Ahmed, Tanbir (2013). "Some More Van der Waerden numbers". Journal of Integer Sequences. 16 (4): 13.4.4. MR 3056628.
  19. ^ a b c d e f g h i j k Brown, T. C. (1974). "Some new van der Waerden numbers (preliminary report)". Notices of the American Mathematical Society. 21: A-432.
  20. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad 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. S2CID 122129059.
  21. ^ Schweitzer, Pascal (2009). Problems of Unknown Complexity, Graph isomorphism and Ramsey theoretic numbers (Ph.D. thesis). U. des Saarlandes.
  22. ^ Karki, Sikij (2024). "A New van der Waerden Number" (PDF). U(t)-Mathazine (9): 34–39.
  23. ^ 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

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