

A171081


Van der Waerden numbers w(3, n).


4



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

3,1


COMMENTS

The twocolor 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. AddisonWesley, 2015, page 5.


LINKS

Table of n, a(n) for n=3..19.
Tanbir Ahmed, Oliver Kullmann, and Hunter Snevily, On the van der Waerden numbers w(2;3,t), arXiv:1102.5433 [math.CO], 20112014; Discrete Applied Math., 174 (2014), 2751.
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.
Sequence in context: A222623 A141469 A046412 * A232056 A109661 A015798
Adjacent sequences: A171078 A171079 A171080 * A171082 A171083 A171084


KEYWORD

nonn,hard,more


AUTHOR

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


EXTENSIONS

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


STATUS

approved



