A171081 Van der Waerden numbers w(3, n).

9, 18, 22, 32, 46, 58, 77, 97, 114, 135, 160, 186, 218, 238, 279, 312, 349

Offset: 3

Author

N. J. A. Sloane, based on an email from Tanbir Ahmed, Sep 07 2010

Keywords

• nonn
• hard
• more

Comments

The two-color van der Waerden number w(3,n) is also denoted as w(2;3,n).

Ahmed et al. give lower bounds for a(20)-a(30) which may in fact be the true values. - N. J. A. Sloane, May 13 2018

B. Green shows that w(3,n) is bounded below by n^b(n), where b(n) = c*(log(n)/ log(log(n)))^(1/3). T. Schoen proves that for large n one has w(3,n) < exp(n^(1 - c)) for some constant c > 0. - Peter Luschny, Feb 03 2021

References

• Knuth, Donald E., Satisfiability, Fascicle 6, volume 4 of The Art of Computer Programming. Addison-Wesley, 2015, page 5.

Links

• Tanbir Ahmed, Oliver Kullmann, and Hunter Snevily, On the van der Waerden numbers w(2;3,t), arXiv:1102.5433 [math.CO], 2011-2014; Discrete Applied Math., 174 (2014), 27-51.
• Ben Green, New lower bounds for van der Waerden numbers, arXiv:2102.01543 [math.CO], Feb. 2021.
• Tomasz Schoen, A subexponential bound for van der Waerden numbers, arXiv:2006.02877 [math.CO], June 2020.

Crossrefs

Cf. A005346 (w(2, n)), A171082, A217235.

Extensions

a(19) from Ahmed et al. - Jonathan Vos Post, Mar 01 2011