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).

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:

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.

• Ramsey number
• Graph coloring

• 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.

• 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.