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	>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

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

as proved by Gowers.[9]

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.[10]

One sometimes also writes w(r; k₁, k₂, ..., k_r) to mean the smallest number w such that any coloring of the integers {1, 2, ..., w} with r colors contains a progression of length k_i 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 ${\displaystyle {\mathcal {E}}^{5}}$ of the Grzegorczyk hierarchy.

See also

• Ramsey number
• Graph coloring

References

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. 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. 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. 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. Heule, MarijnJ (2017). "Avoiding triples in arithmetic progression" (PDF). Journal of Combinatorics. 8 (3): 391–422. doi:10.4310/JOC.2017.v8.n3.a1.
7. 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. 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. Landman, Bruce; Robertson, Aaron; Culver, Clay (2005). "Some New Exact van der Waerden Numbers" (PDF). Integers. 5 (2): A10. MR 2192088.
12. 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. 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. 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. 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. 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. Ahmed, Tanbir (2013). "Some More Van der Waerden numbers". Journal of Integer Sequences. 16 (4): 13.4.4. MR 3056628.
19. Brown, T. C. (1974). "Some new van der Waerden numbers (preliminary report)". Notices of the American Mathematical Society. 21: A-432.
20. 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

• Landman, Bruce M.; Robertson, Aaron (2014). Ramsey Theory on the Integers. Student Mathematical Library. Vol. 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). doi:10.37236/925. MR 2285810.

External links

• O'Bryant, Kevin & Weisstein, Eric W. "Van der Waerden Number". MathWorld.
• Klarreich, Erica (2021-12-15). "Mathematician Hurls Structure and Disorder Into Century-Old Problem". Quanta Magazine.