Lower bounds on some Van der Waerden numbers based on quadratic residues (Q2839726)

The Van der Waerden number \(W(r,k)\) is defined as the smallest positive integer \(N\) such that for any \(r\)-coloring of the set \(\{1,2, \dots, N\}\), there is a monochromatic arithmetic progression of length \(k\).NEWLINENEWLINEObtaining a good upper bound of \(W(r,k)\) is a very difficult problem and currently the best known upper bound is by \textit{W. T. Gowers} [Geom. Funct. Anal. 11, No. 3, 465--588 (2001); erratum ibid. 11, No. 4, 869 (2001; Zbl 1028.11005)].NEWLINENEWLINEFor a prime \(p\), \textit{E. R. Berlekamp} [Can. Math. Bull. 11, 409--414 (1968; Zbl 0169.31905)] established the lower bound \(W(2, p+1) > p 2^p\).NEWLINENEWLINEFollowing Berlekamp's observation, \textit{J. R. Rabung} [Can. Math. Bull. 22, 87--91 (1979; Zbl 0412.10040)] obtained lower bounds for some small Van der Waerden numbers \(W(2,k)\).NEWLINENEWLINEBy improving the efficiency of Rabung's method, in the present paper, lower bounds for \(W(2,k)\) are obtained for next few values of \(k\) (between 11 and 23).