Some results on a class of mixed van der Waerden numbers

Abstract

The mixed van der Waerden number ${w}(k_1, \ldots ,k_r;r)$ is the least positive integer $n$ such that every $r$-coloring of $[1,n]$ admits a monochromatic arithmetic progression of length $k_i$, for at least one $i$. We denote by ${w}_2(k;r)$ the case in which $k_1=\cdots =k_{r-1}=2$ and $k_r=k$. For $k \le r$, we give upper and lower bounds for ${w}_2(k;r)$, also indicating cases when these bounds are achieved. We determine exact values in the cases where $(k,r) \in \{(p,p),(p,p+1),(p+1,p+1)\}$ and give bounds in the cases where $(k,r) \in \{(p,p+2),(p+2,p+2)\}$, for primes $p$. We provide a table of values for the cases $k \le r$ with $3 \le k \le 10$ and for several values of $r$, correcting some known values.