Bounds on some van der Waerden numbers

For positive integers s and k 1 , k 2 , … , k s , the van der Waerden number w ( k 1 , k 2 , … , k s ; s ) is the minimum integer n such that for every s-coloring of set { 1 , 2 , … , n } , with colors 1 , 2 , … , s , there is a k i -term arithmetic progression of color i for some i. We give an asymptotic lower bound for w ( k , m ; 2 ) for fixed m. We include a table of values of w ( k , 3 ; 2 ) that are very close to this lower bound for m = 3 . We also give a lower bound for w ( k , k , … , k ; s ) that slightly improves previously-known bounds. Upper bounds for w ( k , 4 ; 2 ) and w ( 4 , 4 , … , 4 ; s ) are also provided.