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Van der Waerden numbers w(3, n).
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%I #33 May 06 2021 03:21:18

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

%N Van der Waerden numbers w(3, n).

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

%C 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

%C 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

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

%H Tanbir Ahmed, Oliver Kullmann, and Hunter Snevily, <a href="http://arxiv.org/abs/1102.5433">On the van der Waerden numbers w(2;3,t)</a>, arXiv:1102.5433 [math.CO], 2011-2014; Discrete Applied Math., 174 (2014), 27-51.

%H Ben Green, <a href="https://arxiv.org/abs/2102.01543">New lower bounds for van der Waerden numbers</a>, arXiv:2102.01543 [math.CO], Feb. 2021.

%H Tomasz Schoen, <a href="https://arxiv.org/abs/2006.02877">A subexponential bound for van der Waerden numbers</a>, arXiv:2006.02877 [math.CO], June 2020.

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

%K nonn,hard,more

%O 3,1

%A _N. J. A. Sloane_, based on an email from _Tanbir Ahmed_, Sep 07 2010

%E a(19) from Ahmed et al. - _Jonathan Vos Post_, Mar 01 2011