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

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

as proved by Gowers.[8]

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

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 of the Grzegorczyk hierarchy.

See also

References

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  2. 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.
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  4. Kouril, Michal (2012). "Computing the van der Waerden number W(3,4)=293". Integers. 12: A46. MR 3083419.
  5. 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. 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.
  7. Monroe, Daniel (2019). "New Lower Bounds for van der Waerden Numbers Using Distributed Computing". arXiv:1603.03301 [math.CO].
  8. 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. S2CID 124324198.
  9. 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.
  10. Landman, Bruce; Robertson, Aaron; Culver, Clay (2005). "Some New Exact van der Waerden Numbers" (PDF). Integers. 5 (2): A10. MR 2192088.
  11. 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.
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  13. 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.
  14. Kouril, Michal (2015). "Leveraging FPGA clusters for SAT computations". Parallel Computing: On the Road to Exascale: 525–532.
  15. 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.
  16. 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.
  17. Ahmed, Tanbir (2013). "Some More Van der Waerden numbers". Journal of Integer Sequences. 16 (4): 13.4.4. MR 3056628.
  18. Brown, T. C. (1974). "Some new van der Waerden numbers (preliminary report)". Notices of the American Mathematical Society. 21: A-432.
  19. 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.
  20. Schweitzer, Pascal (2009). Problems of Unknown Complexity, Graph isomorphism and Ramsey theoretic numbers (Ph.D. thesis). U. des Saarlandes.
  21. 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.

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