Lower Bounds on some certain van der Waerden Functions

Abstract

For positive integers \(r\) and \(k_1, k_2, \ldots, k_r\), the van der Waerden number \(W(k_1, k_2, \ldots, k_r; r)\) is the minimum integer \(N\) such that whenever the set \(\{1, 2, \ldots, N\}\) is partitioned into \(r\) sets \(S_1, S_2, \ldots, S_r\), there exists a \(k_i\)-term arithmetic progression contained in \(S_i\) for some \(i\). This paper establishes an asymptotic lower bound for \(W(k, m; 2)\) for fixed \(m \geq 3\), improving upon the result of T.C. Brown et al. in [Bounds on some van der Waerden numbers.J. Combin. Theory, Ser.A \(115 (2008), 1304-1309]\). Additionally, we propose lower bounds on certain van der Waerden-like functions.